A scoresheet allows for a number of different things.
1. The scoresheet allows reconstruction of the game.
This is important for several reasons. Imagine that an earthquake occurs and upsets the board. (Or, more prosaically, the table was set up incorrectly and is upset by a light touch.) If there is no record, how would you reconstruct the position? Replaying a game simply because of an upset board is a tremendous waste of time, and it's far more mental strain than a player should be expected to handle.
Additionally, with modern storage and retrieval abilities, each game is seen as something to learn from. Every single recorded game is one that can teach us new things.
(Kriegspiel is an instance where it is vital to record the game for reconstruction. Most of the fun is in the post-game discussion, and it's impossible to do this without recording it.)
2. The scoresheet allows for arbitration of disputes.
Without a record, how would you go about proving to an official that 50 moves have occurred without a capture/pawn move? Without a record, how would you prove that your opponent is a scurrilous lout who secretly replaced his Queen on the board while you were in the restroom?
3. The scoresheet is an aid to players in reconstructing their play during the game.
Top chess players are able to remember large numbers of variations and lines. However, having the scoresheet available reduces one aspect of mental accounting: The player can always look at the scoresheet and mentally reproduce the game up to that point, rather than attempting to remember "now was it exd5?"
4. Signing the scoresheet is a way to ensure that both players accept the outcome of the game.
A player's signature is proof that they are satisfied with what happened. It indicates "yes, I played this match, and it happened this way. I agree that this record is accurate."
FIDE devotes section 8 of the Laws of Chess to the scoresheet. Note particularly that 8.7 states:
At the conclusion of the game both players shall sign both scoresheets, indicating the result of the game. Even if incorrect, this result shall stand, unless the arbiter decides otherwise.