I know exactly what Chrome does and I assume Firefox does the same because I don’t know what else it would do. It basically acts as if you created a temporary profile that’s deleted when you close the window. There’s a little more to it than that, like being able to opt in to using extensions from your main profile, but basically it’s just for things like preventing your family from seeing that you’ve been watching porn or shopping for Christmas presents.
People seem to get confused a lot because it doesn’t use encryption, hide your IP address, or anything like that, and you can still be tracked if you sign into things in the private/incognito window.