Okay, so imagine you really like a video game, like Mario or Sonic. But maybe there are some things you don't like about it, or you wish there were more levels or characters. A fangame is when someone who really loves a video game decides to make their own version of it, with all the stuff they want in it. They might change the levels, make new enemies or bosses, or add new characters that weren't in the original game.
Think of it like building a Lego set. You start with the basic pieces, but then you can add your own pieces to make it unique and different from the instructions.
A fangame isn't an official game from the people who made the original game. It's made by fans who just want to make something that they think is cool. Sometimes people really like fangames and play them all the time, but sometimes the people who made the original game don't like it and can make the person stop making the fangame.
So, it's like a homemade version of a video game that someone makes because they love the original game and want to add their own ideas to it. Cool, huh?