Cause and fix for craftsman lawnmower catching on fire near battery - under seat?

narkeleptik (from comments) is exactly right. The (+) side battery cable grounded to the body of the mower. This caused the leads to go into global thermonuclear meltdown. The "fire" you found is the insulation burning because the wires were hot enough to make them burn.

You should be able to fix this without too much of a hard time. You need to get some wire which has enough girth to provide the amperage for the starter. You'd most likely need to run new power wires from the battery (+) to the starter as well as the ground on the body back to the battery (-). The easy way would be to look on line for Craftsman parts and find the pre-made leads, though this would be more expensive then building your own. I would bet if you check your battery, it still has a charge. It may be a bit low, though, so a stint with the battery charger may be in order.

EDIT: More than likely, the problem was, when the battery was replaced, care was not taken with the positive lead. Two things happened: the insulation on the (+) lead wore through; the paint on the body wore through. When those two things happened, the short occurred allowed unrestricted power flow through the wires, which was more than they could handle, heating them to red hot, burning the insulation and causing the small, but brief fire. Once the lead stopped making contact (was melted into oblivion), the connection was broken. It looks to me the lead was crushed between the battery and the body of the tractor. To tell for sure, look where the (+) lead looks good. Where it ends is where the connection happened.