It's a "Pop Filter". It is used to prevent "Plosives", which are blasts of air that can make a big pop or boom in a microphone. Some pop filters are foam rubber balls or socks that cover the end of the mic, to protect the mic's diaphragm from the air blast. The type in the video is a nylon mesh, and it serves the same purpose. The first prototypes of these were made with an embroidery hoop with a ladie's nylon hose stretched across the hoop. That idea caught on, and now you can buy them like the one in the video, with a flexible gooseneck on it to be able to position it where you want it.
Good pop filters will dissipate the plosives and allow the rest of the sound to pass through. The mesh stops most of the big puff of air formed by plosives.
Plosives happen mostly on the letter "P", but can also happen on "B", "G", "D" or "T" or any similar consonant that suddenly releases a small burst of air. The mic may also have a "Low Frequency" filter switch that cuts off some of the low-end tone of the mic. You also lose some bass-sound from the voice being recorded, and the switches may not be as effective as the screens, so most people seem to like the screens these days. In some old photos of the Beatles, you can see some metal screens being used around the mics.
Many "Ball" mics (hand-held live vocal mics) have a foam ball inside the metal ball-shape on the singing end that serves as a built-in pop filter.