Even if you don't totally understand how the circuit works you can diagnose your problem yourself if you do the following in this EXACT order:
Disconnect the starter solenoid COIL (The smaller two of the four wires on the solenoid).
Turn the key "ON" and hit the button. If it pops the fuse or makes weird zapping sounds the problem is in the wiring UPSTREAM of the solenoid. If that doesn't happen, leave the coil disconnected and move on to the solenoid.
Put the bike in NEUTRAL, key OFF and jump between the big red wires with a screwdriver or the points of some needlenose pliers.
If the starter turns, the solenoid COIL has shorted. You need a new solenoid.
If the starter DOESN'T turn and you hear zapping, popping or humming, the problem is not the solenoid, but between the solenoid and the starter, or in the starter itself. Follow the big starter wire to the starter looking for bare spots, particularly where it passes under and around the front motor mount. Then check the brushes and do the starter checks.
Get a solenoid from a KLR650, or almost any larger bike's solenoid which has higher contact rating. Late model TW solenoids are bovine doodoo, It's a no-brainer upgrade, and probably around the same cost as the stocker.
Incidently the above procedure will get you home if a solenoid fails. Just disconnect the small wires, turn the key "ON" rather than leaving it "OFF" and jump the two big wires til the bike starts.