As paceman says, the screening is held in with a plastic spline. Remove it and the screen will come out. Probably best to get new spline material, along with the screen. Most full line hardware stores will stock it, but splines come in different sizes, so take your old with you for a sample. A spline tool will help greatly with the installation and is probably under $5. I advise you get one.
Can you do it without taking the screen door off? Yes, but it is not really easy. The door is curved and vertical, and holding everything in place while you roll the new spline in is a bit of a challenge. However, since it is very difficult to remove the screen door because of the hinge arrangement, it is the way I would recommend doing it. I would buy extra screen so if you mess up one trial, you have a second piece. It is quite inexpensive.
It can be a little hard to get the screen in tight, and yet not so tight it warps the door itself.
Once you master the technique, you will become the expert to go to for others who need new screens in their doors.
And pat the kitty for me.