How should I reattach this garage door bracket?
Once screws rip out I like to reenforced the door.
I would get a larger piece of metal several inches wider. And screw into the door in 2-3 new locations if in the center use flat head screws there. Then attach the original bracket to the new piece of metal.

Next I would check that the door moves up with only a few pounds of force. I have found garage like you have when the torsion or one of the torsion springs break. If the door feels like more than 30lb you may want to inspect your spring(s). Broken springs are not hard to replace but can be quite dangerous (believe it or not even fatal at times). Door pros can knock a broken spring out quickly. 
This may be the reason the bracket ripped loose.