Okay kiddo, so imagine you have a glass of lemonade and it's got too much sugar in it. What do you do to get rid of some of the sugar? You might add some water to dilute it, right? Well, scientists use a similar trick to separate out the proteins they're interested in.
They take a bunch of proteins mixed together in a solution (kinda like your lemonade), and add a chemical called ammonium sulfate. This is kind of like adding water to your lemonade, except instead of diluting the mixture, it causes some of the proteins to clump together (or precipitate). These clumps of proteins are easier to separate out than a mixture of many different individual proteins.
The reason this works is because ammonium sulfate changes the concentration of the solution. When you have a lot of proteins in a solution, they're all bumping around and interacting with each other. But adding the ammonium sulfate makes the solution more concentrated, which means there's less space for the proteins to move around. Some of them end up sticking together because they're forced into close proximity.
Scientists can then collect these clumps and use other techniques to further purify the individual proteins they want. It's kind of like playing a game of "sorting beans" - if you have a big pile of mixed beans, you might start by separating out the big, obvious ones (like kidney beans), and then go on to separate the smaller ones by size, shape, or color.
So, that's ammonium sulfate precipitation in a nutshell - it's a way to separate out proteins in a mixture so that scientists can study or use them individually.