Open the top of the fuel tank and look inside for low fluid levels. Change the fuel if it is stale -- over a month old -- or there are signs of water in the gasoline.
Touch the spark plug and wiggle it. Tighten it with a socket wrench if it is loose.
Unscrew the clamps holding the fuel line in place using a screwdriver. Clear out the line with compressed air.
Poke a pin through the vents in the gas cap if they are clogged.
Remove the screw from the air filter cover. Inspect the air cover. Throw out a dirty paper filter. Clean a foam filter with liquid dishwashing detergent and warm water. Let the filter dry.
Take the mower to a service professional to have the carburetor adjusted if the backfiring continues. Adjusting the carburetor improperly is terrible for the engine. Also ask the service professional to check the magneto timing, which requires expensive, specialized equipment, and the valves.