My 2-month-old kitten's leg swelled, then it died. What happened?

I am so very sorry to hear about your kitten. If this kitten was outdoors depending on your location I may worry about a snake bite if this came on very suddenly. Other possible causes would include trauma which could cause swelling if the leg was broken and head trauma could cause shaking and death with increased pressure building up in the skull. If you want to find out the cause of death, a necropsy can be performed by a veterinarian. If you decide to pursue that option, it is best to keep the body in the fridge (not freezer) and get it to the veterinarian ASAP.