Sounds like the starter solenoid is clicking, trying to engage but isn't. Providing you can turn the engine screen by hand. Also providing the battery terminals are clean and tight. The best way to test these is to jump across the 2 big terminals, (it will spark) If it cranks then feed power to the small terminal on the solenoid. If all you get is a click, check the ground of the solenoid-then replace the solenoid if it is still inop. The solenoid is where the + battery terminal attaches. To do this test on a rear mounted solenoid, you need to pull the battery and use jumper cables to the battery terminals.

***"Also providing the battery terminals are clean and tight. The best way to test these is to jump across the 2 big terminals, (it will spark)"***
Be careful in what you write! The above suggestions could be taken to mean "jump across the two big battery terminals on the battery". Of course that is not what tomplum is suggesting. And you NEVER want to jump across the terminals or posts on a battery.
He is referring to the two big terminals on the starter solenoid.

If its the single cylinder B/s I had similar problems. It was given to me with mulitiple problems. I figured out it had a bad battery, bad key switch, fuel in the cyliner and needed a valve adjustment.
First I would have the battery load tested.

Good save Mownie... thanks!

***" fuel in the cylinder"***
I hope you have changed the oil!!! Fuel in the cylinder usually means the fuel has also gotten into the crankcase lube oil. You heed to keep a very close eye on this situation as you are likely going to need to add carburetor cleaning to the list of woes. Don't let a leaking carb cost you an engine. Check the oil EVERY time prior to use. If the oil level tends to rise and you can smell gas on the oil, there's your sign.

You new-comers to the lawn tractor field--hear this:
When you pull out the dip-stick to check the oil, wipe it and check it again, because you will get a false reading in the first pull of the stick! This is because the oil that is splashed around from the last time you ran the engine, will show a really high reading! Engines have been blown because folks don't know this fact, and just assume, because there is lots of oil on the stick, that there is lots of oil in the engine.
So--pull stick, wipe it off, then put stick back in the engine, then pull it back out, and you will get the true reading! But, if it still shows way more oil in the engine, then you must look for other troubles. A leaking float in the carburetor will allow fuel to get into the crank-case of the engine, and quickly ruin it--with a big "BANG", and a new ventilation hole in the engine block!
Been there--done that!

I have a had my bronco for 6 years now and ever since I bought it it has backfired when shutting off, no matter what the throttle level is at when it is turned off.

Recently I have had a problem with it cranking and now it won't crank at all. I get the click behind the seat (solenoid). I tried jumping the positive leads to bypass the solenoid and still get the same response. Also, the battery cables are extremely hot at this point. The engine turns, or tries to turn, but no dice.

I am leaning toward fuel in the cylinder or a bad starter. Is this a problem with these engines?
Any thoughts?

Daniel, please post the full model number & serial number of the Bronco from the data sticker on the chassis. Also look on your engine and find the Briggs engine model number & type number.
Knowing these numbers will let us view some manuals & IPLs online so we can make a better determination of the problem.

What makes you think fuel in cylinder?
Have you removed the spark plug (or plugs) to see if the engine spits out fuel when the engine turns?
Remove the spark plugs and turn the engine by hand. You do that by placing both hands on the grass screen found on top of the engine and turn clockwise. Watch for gas to be expelled from the spark plug hole. If gas runs out of spark plug hole, you have several tasks to complete before this engine can be safely run again. Post all the data and we will go from that point.

I am having the same problem (fuel in the cylinder verified by removing the spark plug and seeing gas pour out)

My engine is a 31P777-0293-E2. On the online parts lists I see a fuel solenoid. Could that be the problem? or is it most likely a faulty float. If it is the float, do you just unscrew the bowl from the carb.

So far, no gas in the oil (thankfully)

The problem IS NOT the fuel solenoid. The fuel solenoid only shuts off fuel to the main jet for preventing "afterfire" when shutting down the engine.
Your problem is because the fuel inlet needle valve is not completely sealing off the flow of fuel in your gravity fed fuel system. And this is a common problem.

The "cure" is to install an inline fuel shutoff valve between the tank ank the fuel filter and train yourself to cut off the fuel except when the engine is running.

No fuel in the oil? Guess again! Anytime fuel has leaked into the cylinder enough for you to expel it from the spark plug hole.........some fuel has gotten past the piston rings and into the oil. Only the rarest combination of valve positions and angle of the tractor could keep gas from getting into the crankcase. Best to go ahead and change the oil.

Thanks mownie. I went to the local Tractor Supply and found a B&S fuel cutoff valve for about $8. Put it in. Will see how it goes.

" Also look on your engine and find the Briggs engine model number & type number.Knowing these numbers will let us view some manuals & IPLs online so we can make a better determination of the problem. "

It absolutely blows my mind that people continue to ask for diagnosis without telling WHAT engine they have and even worse, people diagnose the problem without knowing what engine.

Walt Conner