The expression "back out" means to withdraw from a commitment, plan, or agreement. In the sentence, the speaker had initially made plans to go skiing but could not follow through because of an injury. Therefore, "back out" is the correct phrasal verb to describe this withdrawal.
Other options are incorrect:
- "back up" means to support or reverse a vehicle.
- "back of" is not a valid phrasal verb.
- "back on" doesn't fit the sentence grammatically or idiomatically.
Thus, the most appropriate choice is "back out".