What is this plastic basin on the back of my fridge, and does it require any special care?
That is almost certainly the drip pan, and it is intended to catch condensation flowing from the condensation/defrost drains in the fridge/freezer.
This water should just evaporate into the surrounding air over time.
Cleaning it occasionally is not a bad idea, but it is not necessary to 'replace' any of the water you may find in it.