I just bought the game on the GOG website and when I open GOG GALAXY it doesn't show in my Library why is that?.

You should contact GOG support. That's not really a question for here.

Still, I would try exiting GOG Galaxy and starting it again. GOG Galaxy does have some bugs.

Edit: Had noticed more recently, a game took an hour or more to show up after being added. Other words, it was heavily delayed. If it shows success, this is probably going to be the reason.