So various localities have tried to interpret that as meaning 'you have the right to do it if you do X, Y or Z' (such as obtaining a permit in advance, or not being too loud too early in the morning, or not blocking traffic, etc etc etc).
In general, you do not need a permit in the USA to hold a protest or demonstration. But in practice, in many cases police will interfere if they want to, leaving it up to courts to decide (much later) if the police actions were permissible under the law. Having a permit in advance will make such police actions less likely.
Source: IANAL,BYDNTBAFLTCOBCRILTO