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Hello...I have a riding mower that I am trying to replace the battery cables because one of them was broken. As I'm putting the new cable on the mower is trying to start on its own. Any suggestions as to what to look for? Yes the ignition is turned off. Thanks.

Either the starter solenoid is shorted, or you have connected something incorrectly

Wire from starter to right side of solenoid. Left side of solenoid wire goes to battery.

now if you can get the mower to cut the grass as well as start its self you would be a made man lol sorry i know this is no help

Most solenoids have 4 wire posts; 2 large and 2 small. It does not matter which of the 2 large lugs the cables are hooked to as long as one side goes to the battery and the other side goes to the starter. it sounds like you are trying to put put both larger red wires on the same side.
If they are going on opposite sides, then disconnect the small red wire on the S terminal. if it is still trying to start, the internal switching mechanism is stuck in the closed position and the solenoid needs to be replaced.