The breasts start to enlarge during this first stage of puberty due to the effects of the hormone estrogen. The estrogen causes the breast tissue cells to grow and enlarge. Some girls probably have a few more breast tissue cells on one side, or the cells may be a little more sensitive to the effects of the estrogen. Many adult women will notice that one of their breasts is slightly larger than the other, but the difference is usually very slight. (Just as most of us have one foot that is slightly larger than the other). Very rarely, women can have a marked difference in the size of the two breasts that can be uncomfortable or require surgical intervention.
I would reassure your daughter (and yourself!) that it is totally normal to have breast development on only one side at first. It does not mean that she will have a significant difference in the size of her breasts when she grows up. If she wants to start wearing a bra, that is fine. A sport bra may help make the difference between the two sides less noticeable, but it may also put more pressure on the breast, and this could be a little uncomfortable if it is tender. If within six months you do not see development on the other side, you should have her evaluated by her physician.