You need to think of "side" being used in two slightly different ways. In the first the "side" is relative to the person being spoken to being the "middle" and there are two areas "beside" them. The second is assuming there's some other thing that is the "middle" and the person being spoken to is on one "side" of that middle.
"I am at your side" and "I am by your side" are using the first meaning. They mean "beside you". This could be meant literally as in they are physically beside you, or figuratively as in not actually being in the area your left or right but with you in a supporting way. "By" is more likely to be figurative than "at" but either could be meant either way.
"I am on your side" most likely means they see the situation as involving two or more opposed groups, and they are saying that they are in your group. As in "I am on your side of the argument."