Well, my understanding of it is basically this:
Zoe Quinn released a game called Depression Quest. It isn't much a game, per se, but it was a creative venture on her part, and it eventually made it to Steam. This made a lot of people angry. Quinn also happens to be a feminist who was outspoken in the gaming community. Then some dude that she was with released the most douchie, "I'm a heartbroken emo kid" diary blog about how she cheated on him (the dude has since expressed nothing but guilt for doing this). One of the guys that she cheated on him with happened to play her game, and do a write up, albeit, not a review. This caused people to believe that she fucked people for positive spin on her game (which, again, was a pet project of hers and never sold really anything of note, to my knowledge). The backlash toward Quinn by people that were claiming to be gamers and for proper journalism (rape threats, death threats, rape and then murder threats) caused a lot of the gaming community, journalists included, to rally behind Quinn. Insults were slung by both crowds, and my understanding is that even those labeled SJWs (social justice warriors) were guilty at times of slinging insults at gamers that were questioning why anyone would defend Quinn. You had both sides with their poor arguments drowning out the good arguments. Somewhere in all of that, my understanding is the gamergate began, where some people decided that they were going to "take back" gaming. That's about as far as I know, as I ran out of energy after that.
Personally, my opinion is that Zoe Quinn didn't do anything wrong, her attractiveness shouldn't matter, no one should be threatening to rape someone, games journalism is as much of a joke as movie or book journalism, and probably no one should be calling each other white nerd basement dwellers. Also, no one needs to take games back. Last I checked, there hasn't been, nor will there ever be, a short supply of games focused on the interests of young men.
EDIT - I should add that a lot of my understanding of this situation post the Quinn part has been through bits and pieces here and there, so I apologize if I miss the overall message of the movement.