자는

The suggestion that we do X

Takes a 'let's' proposal and puts it in front of a noun so you can talk about it: the message that we should live in the moment, the intention of cutting down on plastic, the idea of talking casually, the talk about staying out of trouble. Followed by 대로 it means 'as they suggested' — going along with what the kids decide. It is the modifier form of 자고, so it always leads into a noun.