I don't know the breakdown of gamer types here on BGG (please post some stats if you have them), but I would guess that the majority are light euro gamers. But wherever you start, you are likely to be exposed to wargaming at some point. You might see an interesting image. You might play a game that has a simple conflict mechanism. You might check out a game that your friends or GeekBuddies have been raving about. I'll guess (again) that the average first reaction is negative. At least, that's the sense I get from being on BGG 27 hours a day.
Of the top 50 games on BGG, Twilight Struggle, War of the Ring (first edition), Commands & Colors: Ancients, Paths of Glory, Hannibal: Rome vs. Carthage, Combat Commander: Europe, War of the Ring Collector's Edition, Twilight Imperium (Third Edition), Here I Stand, Conflict of Heroes: Awakening the Bear! – Russia 1941-42, Advanced Squad Leader, and Memoir '44 are marked as wargames, with many others possessing some limited forms of direct conflict.
As I have gotten deeper into gaming in general, I have gotten deeper into wargames. It's partly because I have naturally branched out from my humble euro beginnings, partly because I look into just about anything that I stumble onto, and partly because I am always looking for deeper and richer gaming experiences. If you want to get heavier (not just more complicated) than Through the Ages: A Story of Civilization, for example, there's really no better place to go other than wargames.
However, wargames are difficult to get on the table, and virtually impossible to even introduce to non-wargamers without having to push a little. The following are the barriers to entry--perceived or real--that I see for wargames in general, and my reactions to those. You can always find examples of wargames (and others) that do not exhibit the typical traits I will discuss, but I'm going to ignore the extremes.
Wargames have long rules: They do. If you want to play a wargame, you are going to have to expect to put more effort into understanding more concepts than your average euro. But you'd be surprised how much overlap there is in concepts between many wargames, especially those of the same scale (tactical, operational, etc). For example, many tactical games have line-of-sight (LOS), and many operational games have zone-of-control (ZOC).
Wargames don't use conversational rules style / Wargame rules aren't written for learning the game: Many wargames have rules numbered like 3.10.5. With more concepts comes the need for more organization. First of all, it's much better to refer to rule 3.10.5 when talking about it instead of something like "page 6, second column, third paragraph". This system also avoids the issues that would occur if two people used two different printings of the rulebook (numbered wargame rules ALWAYS keep the same rule numbers). Secondly, wargames--more than any other games--have systems with a lot of conceptual feedback. For example, you cannot talk about movement without talking about defensive fire, and vice versa. And in a related manner, wargames use the same concepts in multiple places. For example, notions like fire attacks or supply lines might be used in many places in the sequence of play. Having all the rules for one concept in one place is really a good way to go.
Wargames have too much chrome: I suppose it depends on your definition. If a game has, say, infantry and cavalry, do you want them just to be generic units with perhaps a different strength or movement rate? Or do you want them to have some semblance of the properties of their real-world counterparts? It might come down to what the game is trying to represent as well as its scale. If the differences are important to the game, many different properties are justified. This might affect the values associated with each unit, special cases based on terrain, etc. If everything was a cube, there might not be any "game".
Wargames have terrible components: I love my wooden bits as much as the next euro gamer, but they rarely work well in the context of a wargame. In order to differentiate units in a significant way, many values must be present. The "chit" is the most natural and functional solution. It's really no different from Race for the Galaxy: put a bunch of icons/values in predefined positions, and use them in clear and consistent ways. In Race, a new card could be introduced to the game, and you would instantly know all of its properties at a glance. The same holds true for chits. For example, in ASL, being able to differentiate the subtleties between a German "5(2)-4-8 E" and a German "4-6-7 1" is vital. Replacing these units with figures doesn't work. You'd need to be able to differentiate dozens of figure types on a map, then lookup their stats on a table.
What about paper maps? There are several practical reasons why paper maps are preferred by some: cost, weight, smaller boxes, no seams. Some of the more popular games are getting a more deluxe mounted board treatment (eg Twilight Struggle), and some older games are being reprinted on cardstock (eg ASL). The artwork tends to be less busy than a standard euro with professional art. This helps to visually distinguish the board from the large number of playing pieces on it.
What about hexes? A hex grid is the best way to uniformly break up an area to allow for discrete movement in "arbitrary" directions. Any form of grids is going to introduce "rounding errors" in movement, but I find hexes to be a very good playable abstraction. Of course, not all wargames use hexes. There are maps with countries or other variably-shaped regions, point-to-point maps, etc.
Wargames take too long to play: As with any genre, there are many different lengths. However, once you become immersed in a good wargame, you will find that an hour is not enough to play out a scenario with any amount of detail. Games that end sooner than expected might be disappointing. You might find that spending an entire evening engrossed in a good wargame is more fulfilling than 3-4 light euros. Wargames tend to NOT feel at all like "rinse and repeat" for hours and hours. There are discernible arcs to the narrative. You will find yourself thinking a lot about the game you just played.
I'm not a history buff: Neither am I. I know that WWI/II happened in the 20th century and that the Roman Empire was somewhere east of me. That doesn't spoil my fun at all. Wargames pose challenges of many types: spatial (position, range, connectivity, facing/flanking), unit management, timing/sequencing, bluffing, usage of terrain/elevation/cover, command & control, tactics/strategy, politics, negotiation, etc. And you just might find yourself learning something along the way.
Wargames are all the same: Perhaps in some very superficial way, yes. Wargames are about conflicts. But they are as different (or more so) than the entire gamut of euro games.
There are differences in scale. Some games have individual soldiers fighting over very short distances. Some have platoons, battalions, corps, divisions, or entire armies. The entire way you think about playing these games, as well as what factors matter, changes with the scale. How far away can units engage? How does terrain/elevation affect movement and combat? Are turns long enough that supply is a factor? Are they long enough that you can consider "making more units"?
There are different map styles: square, hex, area, point-to-point, etc. The way that "locations" are laid out and connected to each other affects the game play. Gridded maps allow for open movement. Area maps typically conform to geographic and/or political boundaries. Point-to-point maps typically align with meaningful locations (eg cities) and travel between them (eg roads, passes, straits, sea).
There are differences in goals. Some games are about attrition (eg killing so many units, or specific units). Some are about control (eg occupation of a building, or a city, or a country). Some are about getting units across a map. Some are about VP's (generically scoring points for various accomplishments). In some games, each player has different goals, or the goals change over time.
There are differences in implementation. On one end, a wargame can be all about design for simulation. And as I've said before, simulation means "a credible pretense of reality", and not some higher purpose. These games let the players make choices in detail, dealing directly with the systems of the conflict itself. On the other end, a wargame can be all about design for effect. These games step back a bit from the details. The goal (maybe) is for the results to be correct, but the steps the players execute to get those results may not seem "realistic".
The wargame decision space is too open: Consider a game like Ra. On your turn, you have two choices: draw a tile, or invoke Ra. Even if you have no idea what's best, the choices are obvious. In a wargame, you might have dozens of units, each with different characteristics, positions, facings, etc. Combine this with all of the possible interactions with every other unit on the board, and the choices seem almost limitless. I suppose it's analogous to a person who has never played an RPG. The GM doesn't ask, "Do you want to move or attack?" The GM simply inquires, "What do you want to do?" It IS wide open, and that's a good thing. You can make plans, attempt to execute them, and adapt to changing circumstances.
Wargames are too random: Say you are into archery. You fire 100 arrows at a target. 5 hit the bullseye, 12 hit the red, 15 hit the black, 24 hit the white, and 44 miss the target completely. Now you want to "simulate" the act of shooting an arrow at that target with your skill level. You've already figured out the answer, right? Roll some dice, and lookup the answer in a table. That's exactly what many wargames do. The tables are designed to produce an effect that makes the game work. If a game had 100% predictable results for every future action, it would be an abstract (eg Chess).
That's not to say that all wargames handle uncertainty in the same way. Some use "buckets of dice" to inflict hits. Some use a small number of dice and lookup the results on a table. Some use cards. Some have no randomness at all, but use hidden information in the form of blocks to keep things interesting.
Also, "results" are not always "loss of hit points". Units may be pinned, broken, suppressed, routed, reduced, demoralized, etc. A unit/stack may be able to abort its attack or retreat to negate some "damage".
Wargames are not as fun as XYZ: I can't help you there. You like what you like. But I would urge you to not fall prey to preconceptions of wargames. And if you "played one" long ago and hated it (eg Risk, Axis & Allies), do not assume that all wargames are the same (above). In fact, there are more wargame systems than there are euro mechanisms. I'm sure there's one for almost everyone.
I have a moral objection to war/conflict: In Through the Ages, you can attack each other. In Dominion, you can steal. In Battlestar Galactica, you can airlock a player. In Endeavor, you can keep slaves. In Dungeon Lords, everyone plays the bad guys. In all cases, it's just moving bits around. Games use theme to tell a story and/or to map the system to something real so it's easier to remember. Wargames are no different, except maybe that they represent more "real" things than other game types. Wargames don't glorify killing; they let players relive events, learn, understand, experiment.
You've probably already tested the waters: If you've played Through the Ages or Tigris & Euphrates, you've already experienced the idea of directly attacking an opponent to gain military, material, or scoring advantages. If you've played Power Grid or any of the railroad games, you've already dealt with connectivity. If you've played El Grande or Die Macher, you understand the notion of fighting for control. If you've played Goa or Saint Petersburg, you have had to manage your economy. If you've played Race for the Galaxy or 7 Wonders, you've encountered hand management. If you've played Chinatown or Cosmic Encounter, you've tried to trade/negotiate with opponents.
The deep end of the pool: I'm not a fan of the concept of "gateway games". If you have any interest whatsoever, I think you should just dive right in and spend the time getting over the initial hump of some relatively meaty wargame. If you aren't willing to read rules for a few hours (knowing you will not get it all, and that you will make mistakes the first time you play), and spend a couple of game nights just playing a single game a couple of times, then the entire genre is probably not for you. There are certainly many shorter, simpler wargames out there. But I think you may not get the right impression from such a game. That's why I would never recommend Ticket to Ride or Settlers to anyone interested in gaming in general; they give a false impression, and teach all the wrong things.
Wargaming is active on BGG: Wargamers are passionate about their hobby, maybe even more so than euro gamers. The Wargames Forum is the most active subdomain forum on BGG. The discussions are very enjoyable, and the people are helpful.