How to fix bottom screen

You have a bit of an odd setup here. This appears to be a kind of commercial door, but I've never seen a commercial door with screening before. It's very peculiar.

That being said, from experience working with commercial doors I believe there should be a snap in molding holding the screen in the door. Theoretically, this would be similar to how commercial glass is held in place.

Your next step is to check to see if you can find which side of the trim comes off. To be honest, if you've never worked with a commercial door before then I would highly recommend contacting a local professional to come out and take a look at it for you. This could turn out to be a complicated project due to how non-standard this is.