What does that screen do in front of a recording microphone?

It's called a pop filter. It can be used to protect your microphone from spit, but it's main use is to remove plosions. When you get up close to a microphone and talk, when you use "p" sounds, "f" sounds, "s" sounds, and things of the sorts, you get a popping, crackling, or blowing sound. A pop filter just reduces that drastically.