...
What's the Point of Draws?

The point of agreeing to draws if you are bored with the game and/or think that you ill end up in a draw anyway. (Or if you think you'll lose)

you can't choose to win you can only try, i encourage you to play against a strong chess engine and "choose" to win and see how well that works out for you.

But otherwise..........WHY!
If the position is such that a checkmate is very unlikely/impossible then agreeing to a draw is the only sane option. If you can draw against a higher rated opponent your ranking will improve.
At the high end/professional level of the game draws are very common, even with lots of pieces still on the board, when both players realise the position is such that neither side will be able to make a win.
A draw is better than losing. In some cases it isn't possible to win, so you play for a stalemate, trade off to insufficient material or what not. In other cases both sides feel that the play has little to offer to either side, and agree to a draw. As you get better you'll start to get more of a feel for this. At the 800 level, you may need a bit of growing first to see this.
Consider this endgame. You have a king and a bishop. Your opponent has a king and a pawn. Your only chance to NOT lose is to draw (excluding a rare circumstance and idiocy on your opponent's side). The reason behind this is once you take the pawn you have a king and bishop vs a king, insufficient mating material. But if you don't take the pawn and it promotes, a king and bishop vs a king and queen game is lost for you.
Another thing to consider is you opponent in the aforementioned scenario. They were likely down a piece going in to such a scenario. For them forcing you to draw was averting the worst case scenario, losing. To draw from a losing position is a victory in itself.
Another situation to consider is a tournament. You're up in the top 3. The game is going badly and the opponent is higher rated and better than you. You could try and play it out, but you spot a draw by repitition. If you take it you'll have to settle for a draw, but you won't risk the loss.
All this said I tend to play aggressively and avoid drawish situations when possible, but sometimes in a well played game the draw is unavoidable.

My opponent has spent the last 40 moves beating the tar out of me. Now he has me exactly where where he wants me, moves in for the kill, when all of a sudden, I uncork the blinding tactics he overlooked, and......Stalemate! The game is a draw!
The Draw gives a player a reason to play on, even after he's down a rook.

Draws can be boring. In Reykjavík Open 2013 the strong GM's on first board draw in second move in the last round! 1.e4-e5 2.Nf3-draw! They shared the winning price.
Nobody wins a game of chess. The game is lost because the loser made an error. If nobody makes a mistake, the game is a draw, and both sides can claim to have played an errorless game.

You could continiue here as white, but your opponent has to make a very big mistake for losing. So most players would agree to draw (2 of my friends has won this endgame, so it's not impossible to win against inferior play).
It would not be a mistake for white to agree to a draw here. With perfect play the game is obviously drawn, and most players can draw this blindfold with black. So there is really no reason to play for white. Some players would play on, to see if black know the idea (One of my friends played a rapid-chess tournament with 7 rounds, and used at least 30 minutes in 3 games with the black pieces to show that even with time pressure, he could draw this), but there really is no reason.
Why would you want to have a draw? Wouldn't you want to just beat the other person and get a higher rating?