You'll need pet resistant insect screening. I forget which one I used but it works really well. Holds up against cats clawing on it. I remember when I bought it there were two major brands but now it looks like St Gobain bought New York Wire according to this http://catproofwindowscreen.blogspot.com/p/new-york-wire-pet-d-fence.html
You'll need spline. It comes in different sizes but I'm not sure it's that critical or I got lucky when I ordered mine and got the right size. A spline roller helps too.
The screen you have already looks like the cat proof window screen and it doesn't appear to be ripped. You might want to see if you can get the screen back in the groove, then press the spline back in place. The back of a utility knife or some other rounded shape works. Your spline may be too thin so try ordering a larger size spline?
It might be hard to get it back in place. Normally you cut the screen larger than the opening, stretch it in place then insert the spline before you trim out the excess. Might not be enough to work with but worth a shot to try and save some money.