Why do player start with fighting for the corners in go?
Common starting practices are to open by placing stones on a Hoshi (4-4 point), or close to it, in order to "claim" that corner. As the players are trying to secure as much area as they can, it makes sense to start out in a place that is easily defended, and that is something that's true for corners more than for any other location on the board at the start of the game.

Taking control of the center is much easier if the player has a solid base or even a number of solitary stones already placed, which obviously isn't the case at the start of the game. Furthermore, by trying to gain control of the center, your opponent would likely take all or most of the corners for themselves, which will eventually result in having to defend the center against all flanks.

In a real game, of course none of this will be this clearly separated, but those are the essentials it comes down to. Generally, you're not looking to immediately engage with your opponent, but rather to define areas of advantage for yourself first.

Because this question, just like most questions about Go, pose a larger complexity than would be feasible to go into here, I'll leave it at that and provide a few links that are likely to be reachable in the future to read further:


First moves
Open in the corner
Why don't we open at tengen




It should be noted that this is true for 19x19 boards - on 9x9 boards, you will much more commonly see players starting out at the Tengen (center starpoint), or at the very least try to compete for it much earlier on, simply because there isn't as much area to gain in the corners by comparison, and approaching the same strategy as on a 19x19 board would have you run into your opponent much faster than you'd like.