What is this microwave filter thing?
It is an exhaust filter, primarily intended for if the stove fan exhausts back into the kitchen instead of to outside the house.

This filter has charcoal inside to absorb odors so they don't get recirculated through the kitchen.

For your GE model jvm3160rf3ss, its part number is WB02X11550. The product page there has looks to videos that show where this part goes and how to replace it.