Why does a riding lawnmower backfires when you turn it off and how can you stop it?

That is caused by too much gas getting into the engine after the key is turned off. If the mower is new enough to have a anti-backfire solenoid on the bottom of the carb then it must not be working. If it is an older mower and doesn't have one then the carb needs adjusting to back off the amount of gas.