Why would my sprained finger be hot and swollen?

Inflammation . The body's reaction to injury is inflammation. Inflammation has four symptoms - pain, redness, swelling and heat.
Inflammation. Any time you sustain an injury, your body mounts an inflammatory response to try and heal the injury. This response is characterized by warmth, swelling, and redness. Sometimes this response is do robust (ex. Multiple injuries in a car accident) that it can actually kill you by causing multi-organ failure.