Bird's nest custody is like when mommy and daddy birds take turns sitting on the eggs and taking care of the baby birds. Instead of the baby birds going with mommy bird or daddy bird when they fly away, they stay in the nest together and the mommy bird and daddy bird take turns visiting them.
In people terms, it's when a mommy and a daddy are not living together anymore, but they want to make sure their kids continue to live in the same home. So, instead of the kids moving back and forth between two different homes, the mommy and daddy take turns living in the home with the kids.
For example, maybe the mommy lives in the family home with the kids for one week while the daddy lives somewhere else, and then the next week the daddy moves into the family home with the kids while the mommy lives somewhere else. This way, the kids get to have a stable home and they don't have to worry about moving back and forth all the time.