It sounds a lot like "vragen" (to ask in Dutch), and when someone asks you something, you answer. :D
Here's a reconstruction of the root to all of the words mentioned in this thread!
Whereas the Irish would correspond to something like "worthcare" or "weirdcare", and has the same roots as the phrase "versus garrulous". [the frea- comes from a word meaning "turn", and the -graímid comes from a word meaning "call out, shout". In Germanic, the former gives us words relating to how things "turn out" (German "werden"), and the latter gives us English "care", from the sense of crying out as sorrow or grief.]
According to wiktionary 'freagair' comes from 'fri-' (against) + 'gair' (to call), so it literally should mean respond. On the other hand, it should be noted that the English verb 'answer' is also formed in a similar manner - from 'and-' (across) + 'swear' (declare), so it's just a matter of technicality here.
I was not aware that there was rolling R's in Irish. When she pronounced Freagraím, the second R was a rolling R. However, in Freagraímid, the second R sounded more like the other R's. Can someone with knowledge in pronounciation explain what's the right way to say it, and if there actually are rolling R's in irish?