When animals live in packs, one animal is usually the dominant leader, while the others fall into more submissive roles. To be submissive is to obey or yield to someone else.
When you are submissive, you submit to someone else's will, which literally, you put your own desires lower than theirs. You can see this in the Latin root of submit, submittere, which is formed by sub- "under" + mittere "send, put."