What is the importance of the game scoresheet?

In general, the scoresheet is the official record of the result of the game. In most sports I've played, it is signed by the captain (or other representative) of each team at the end of the match to show that they agree with the final score, and then sent to the organising body to show the result of the match (and other secondary things like any disciplinary sanctions applied to players).

To quote from the offical volleyball rules (just as that's the sport I'm probably most familiar with):

AT THE END OF THE MATCH, the team captain: thanks the referees and signs the score sheet to ratify the result

I'd imagine similar things exist for other sports.