I do not have much experience with home improvement related things. I moved into my first apartment and I saw that the internet coaxial cable was pushed inside the outlet. I loosened the screws and using a knife I cut the thin paint layer around the cover and removed the cover
Low voltage plates without brackets make for a sad DIYer
It appears the last installer tried simply screwing the cover plate to the drywall directly without a bracket or box to back it up. Unfortunately for them, screws don't hold well in drywall, as you found out the hard way. While you could use drywall anchors, there isn't much left there for them to hold on to, thanks to the screw-job chewing up the drywall.
It'd be better if you got something called an "old work low voltage bracket", such as the one shown below, that installs behind the drywall with a set of built-in clamps that hold it to the wall. You'd have to cut out the damaged drywall around the existing hole to the size of the bracket's cutout, insert the bracket, and then engage the clamps. Then you can screw the faceplate to the bracket and have things stay together properly. (Obviously, you need to undo the cable to do this.)
Use butterfly plugs or drywall Fisher plugs