To "do someone's legs" is to undermine someone to the point where they have lost their credibility. It can be something like pointing out gaping holes in their argument so their position becomes untenable.

Getting "a bit gobby" in this instance is someone who is somewhat overconfident and is trying to make themselves sound cleverer than they actually are. It's derived from "gob" - the slang term for a mouth - and can also mean someone who talks non-stop, or someone who adopts controversial viewpoints in inappropriate social situations just to get a reaction out of folk.