Answer:
The British fired first, but were forced to retreat when the colonists fired back.
In truth, the British opened fire first, but merely to scare the colonists away while they were still on horseback. The colonists had no means of knowing this until it was too late; all they knew was that something horrible was happening and that they needed to defend themselves.
In either camp, there was no battle, no casualties, and no victory. However, the incident served its purpose: it provided a justification for the colonists to take up arms against Britain. The British had made their intentions obvious by shooting first: they wanted to fight.
You may be wondering why the British would want to battle for something like this. The answer is politics. England was ruled at the time by a king with little control over his ministers. He could pass legislation, but not everyone agreed with him. So there were people in Parliament who wanted more authority - power over the military, law enforcement, and the colonies - and they used this situation to advocate for change.
Due to the fact that several states celebrated their independence on July 4th, this episode has been dubbed "America's Birthday."
Hope this helps! :)