First, this is not a phenomenon that is unique to the US. Protestors in many places, including the UK, are often arrested or otherwise controlled by police. Protestors in the US are often left alone.

But because you asked about the US specifically, the US Constitution protects a right of freedom of assembly, but this has essentially never been interpreted as a license for large groups of people to do whatever they want. Police will often monitor protests and order them to disperse if the crowd becomes particularly agitated or disruptive. If protestors don't disperse (or actively resist police control), police might starting making arrests.

Where exactly this line should be drawn is enormously controversial and contentious, and both police and protestors have been known to behave poorly. What happens can depend on the issue at hand, the demographics of the police and protestors, and the general political climate.