I'm slightly confused about the license. If I use this in my game, or some of it in my game, does that mean I have to make my game open source or make the code public?
Not at all! MIT license just requires that you place a copy of this license in a visible place somewhere (as in, “I used X code, here is the license”). Godot engine itself is also MIT license, so this is a requirement you have to fulfill when releasing a Godot engine game regardless.
Credits are a good place to dump this - self plug but I did this in one of my games if you want to see an example.