See the person next to you? Kill! Kill!!! KILLL!!!!
I'm your baby! Can you feel it?
On the old board I had, they marked that centre area, if you placed a piece outside there your opponent could get an edge placement.
Exactly. They mark the weak spot. If you place one piece in the inner square of the dot, you are offering your opponent that corner, which is an extremely strong piece, since it can never be reverted into a colour of your own.
The outer rim squares are generally winning positions (good othello players know how to turn things around), but the corners are just death sentences most of the time. The dots help you remind that.