To brace one's self means to prepare one's self due to apprehension. The word "brace" is found in "braces". Braces in dentistry are a tool used by orthodontists to straighten someone's teeth. This joke is playing on the word "brace" because it can be found in "braces" and the joke is referring to an orthodontist visit.
For the orthodontist visit , the boy had to brace himself .