The answer is that fire departments respond to any call where there could be an injury or other related need, because they are also EMTs, and in the U.S. you just call 911 and tell the operator what's wrong. If you say you want an ambulance, chances are, you'll get an ambulance... and a fire truck. This is partly due to the fact that most ambulances are a private service and don't really represent the service of the city (which firefighters / fire EMTs do represent).
The problem is that we don't really have a sort of emergency first responder triage department. The operators are it. There's isn't an EMT who can speed to the scene in a fuel efficient car on his own to bring you some bandaids or water and check your blood pressure (e.g., in the case of a homeless person prone with a needle in his arm), and then call for more help if needed, so, instead, a $300,000 hook and ladder truck full of 4 highly trained firefighters (also trained as EMT) is deployed to do the same thing.