Their remote monitoring stations report the data to their servers typically in 15 minute intervals.
If you're worried about the water level at a particular location, they have a service which allows you to get alerts if it exceeds a certain level. They're in the middle of redoing many of their online tools, so it's a mix of legacy and modern looking web development.
They get their funding from contracts with other federal agencies or private businesses, not directly from the government. For example, a company may have a permit that they can pull 100,000 cubic feet of water from a stream per day, but that the stream level must not go below 5 feet. USGS would add monitoring after the draw point and provide that data to the client (and public) at the client's expense.