Ran fine. shut off. will not start. tries but backfires

It could be as simple as a sheared flywheel key and/or the valves needing adjustment. The OHV engines that come in lawn and garden equipment these days are notorious for being very picky about the correct valve settings, especially single cylinder engines. Another common problem involves the fuel inlet valve in the carburetor. Not the solenoid valve but the float valve inside - if it doesn't seat properly fuel will overflow the carburetor and enter the engine. Several bad things happen after that - the cylinder can get enough gasoline in it to lock the piston when you try to start it and that can result in the aforementioned sheared flywheel key. The gasoline will also get past the piston and into the crankcase where it dilutes the oil and the result is poor lubrication. one of the first components to suffer from the lack of lubrication is the camshaft, particularly the exhaust valve lobe, which wears down to the point that it does not open the exhaust valve far enough and the exhaust cannot be expelled fully on the exhaust stroke. The leftover gasses then escape from the engine through the carburetor when the intake valve opens, sounding like a backfire but rather than fire coming out the carb you get popping and raw gas blowing out the intake. Check your oil for the odor of fuel. If it smells like gasoline and, perhaps is inexplicably overfilled with oil according to the dipstick, plan on having the carburetor inlet needle and seat replaced, possibly the flywheel key replaced and probably the camshaft replaced. Keep your fingers crossed that it is just the valves needing adjustment.

Remember the acronym KISS with all due respect (keep it simple stupid) as a long time turf equipment technician I and my coworkers always try to comlicate matters when usually the awnser is simple. I would recommed you start with the spark plug! I have seen spark plugs cause some very unusual syptoms.