The large red wire connects to the solenoid. There is another connection to the + terminal of the ignition switch. The solenoid is an electric relay that transmits high-current battery power to the starter. The transmission is controlled by a low-current connection to the ignition switch.
When you turn the ignition switch to "start", power is applied to the solenoid control terminals, the solenoid engages, and high-current power is applied to the starter. The starter then engages and the motor spins. If the ignition switch could handle the high current needed to spin the starter a solenoid wouldn't be needed. The solenoid enables a low current switch to control a high current starter. The current levels through the starter would incinerate the ignition switch.
You don't have to disconnect anything. Simply connect the red jumper cable between the positive battery terminal and the starter lug (the connection where the big red wire connects to the starter). Then connect the black jumper cable to the tractor chassis or engine block. Then connect the other end of the black jumper to the negative battery terminal. When you make the last connect there will be some arcing. The starter should spin.
If you're not comfortable doing this you may want to consider taking the mower to a repair shop. Not everybody is a mechanic.
