The punchline of this one seems to be that everyone calls her “brat” because of the fact that she’s been using a scalpel since early childhood but most people assume that’s it’s just because of her attitude. I’m sorry if I don’t explain it well
Scalpel in Japanese is mesu. She's a child (gaki) who uses a scalpel, so mesu + gaki = mesugaki. Japanese has a lot of homonyms, so puns are a popular form of humor for manga authors and a challenge for translators.