What do I do to the remaining part of the branch for the shrub where the branch tore off?
It really depends on the kind of shrub it is (do a search on gardeningknowhow.com if you know exactly what kind of shrub it is), but these articles will give you some general guidelines when it comes to cutting plants back:
You can clean up the broken edge with a clean sharp trim if possible.
The best thing to do is let the edge heal on it's own.
Watch for any signs of disease or pests.
Neem Oil can be used to treat the plant if necessary.