Where did the term coming out of the closet originate from?

The closet is where you hide things that you don't want others to see. There's also the idea from the phrase, "Skeletons in the closet" meaning you've got something to hide.

Gay people would often try and hide the fact that they were gay. If they didn't tell anyone they were still "in the closet".

During the Gay Liberation rallies a common cry from the marchers would be "Out of the closets and into the streets!"