So, someone else came over, signed into their account. I try to open GMod, and it asks me if I want to buy or ask permission, but I've already bought it. I've signed out the other person and it still asks the question. Any suggestions?
You could use family share, by going to your username up at the top right, clicking it and going to Account Details.
From here you can click Family Share, and manage all the people who can use your account.
You can ask from permission from your friend and they will have to go through some short steps to allow you to play their games on your computer.
Just because a game is installed on your computer doesn't mean that it is in your Steam Library, ready to play. So you should use family share to do this.
HINT If you use family share, you two can't play the game at the same time.