This looks like the F-N-R selector on the fender and the speed selector on the dash. Now see if this related. You were driving the mower in a higher speed when you pulled into the shed. You moved the direction selector back to N but left the speed selector at the higher speed. You then release the brake/clutch pedal and then turned the mower off. The the next time you got on the mower You pushed the brake/clutch pedal down and started the mower, and moved the speed selector to a slower speed and put the mower in reverse and it wouldn't move. if this is correct then start the mower and and with the direction selector in N put the speed selector at a high gear speed and release the brake/clutch pedal and then push the pedal back down and move the speed selector to a slow speed and then put in either F or R and see if it will move.
What I am describing is the fact that the front drive belt is set for high speed operation at shutdown if the pedal is released with the engine running with the speed selector set for high speed and then shut off, and the next time it is started and the speed selector is moved to a slower speed the belts are synced for high speed operation preventing engagement for low speed. The sequence that I described resyncs the belts for operation.