Make sure the battery on the mower is charged and is able to hold a charge(The battery may just be really shot). Then, just hook up the red jumper to the battery (you can leave the black disconnected, it really doesn't matter, there is already ground between the battery and starter through the frame of the mower). Then touch the other end of the red jumper to the terminal where there should already be a red cable bolted onto the starter. This will bypass any possible faulty wiring or a faulty solenoid, this test will mimick what happens when you turn the key to start it. If it cranks the engine, then its good, maybe you just need a new solenoid, or maybe its bad wiring somewhere. If the starter just spins then the "Bendix" assembly on top is faulty OR the gear on top of the starter is stripped as it is usually made out of a plastic material that wears out over time. You can usually order a Bendix rebuild kit for any starter instead of having to replace the entire unit. This way is usually cheaper. If you don't want to fuss with that, then just buy a new starter. Just beware starters are not that cheap.
Reply when you have tested the starter, i can probably help you further.