I'm sure you can find more detailed answers on any of 100 other forum posts on the topic, but the short answer is just that one core set does not give you enough cards to build 4 legal decks. Once you get a few expansions, you could play 4p with one core.
This is totally wrong. You can play with one core set...but FFG asks you to buy two cores because they can't allow 4 persons to enjoy from a 40$ purchase. In some states (and other countries like Japan) it is illegal to have a night of fun for only 10$ per person.
On a practical level...once you purchase the second core you get an online code to redeem which allows you to use the first core with 4 players. Before then the cards are blurry and cannot be used.