Well, you'd start by wiping them down with wet paper towels to get all the mud off. The flexible parts (sides, arches) can simply be brush polished. That's just using a circular brush to apply a thin coat of polish in a circular motion. You apply it to one boot, then let it sit while you do the other. By the time you finish the second boot, the first one is ready to be brush-polished. You just use that big horse hair brush that comes in the kits and brush it all off until it starts to shine. Use polish on the parts of the soles that are showing also. Repeat as necessary until they're completely black.

The toe and heel has to be spit-shined. You do that with terry cloth and water or window cleaner (spit works but leaves a cloudy sheen). Just wrap the cloth tightly over your first and middle fingers and tuck the excess under your thumb so it stays nice and tight. If your boots are in this bad of shape, you're going to be scooping out some hefty globs of polish to start with. You'll want to rub it softly so it fills in the cracks and dings, then once the polish starts to set you rub a little harder. Again, repeat as needed. Same with the heel.

Avoid liquid polish and spray paint. Yes, I know someone who Krylon'd their boots.