Summarize:

Using a straightedge or sharp knife, cut a clean hole around the screen tear. Make this new hole as small as possible to keep it manageable. Leave at least 1⁄2–1 inch (1.3–2.5 cm) of screen between the hole and the window frame. The new patch should be slightly larger than the rectangular hole. Measure the patch before you cut to make certain it is at least  1 inch (2.5 cm) larger than the damaged area. Slack ends around the opening will make it more receptive and adhered to the new patch. Make sure the entire perimeter of the patch is loosened as well. Bend each unraveled end on the sides of the patch at a 90 degree angle. Work the patch ends through the weaving surrounding the opening in the screen. Then, bend the wires of the patch flat on the other side of the screen to hold the patch in place. Finally, adhere the patch to the screen with clear, waterproof silicone glue.   Wipe away glue drips with a microfiber cloth before they harden to keep the repair smooth. Alternatively, purchase adhesive-backed patches to securely press over the damaged area without glue.
Tim the damaged area into a neat square or rectangle. Cut a piece of patch screen material for the damaged area. Loosen the perimeter weaving of the opening and the patch. Feed the bent patch ends through the screen.