There are a few ways to do it, but the most common is probably to use a tool called a stripper. It looks like a long metal pole with a blade on one end and a claw on the other. You use the stripper to remove the old screen by pulling it off of the frame. Be careful not to damage the frame while you're doing it.

First, remove all of the screws or nails that hold the screen in place. Second, remove the old screen material. Third, replace the screen material and attach it using screws or nails. Fourth, replace any trim or molding that was removed in Step 2. Fifth, reattach anything that was removed in Step 1. Finally, enjoy your newly rescreened patio!