Docilely is how you do something when you are being docile: quietly, without a fight, doing as you are asked. A child might sit down docilely. A horse might follow docilely. The -ly just turns the adjective into a word about the way someone acts. It is not a word people say every day; quietly or without a fuss is more natural in speech. When you do meet it, it is often in stories, and it can sound a little cold, as if the writer is watching someone give in.