From a Percy French song, “The Pride of Petravore” context
What did Big McGrath mean exactly? I get that it is some king flirty banter that won her over, but what exactly does “I know a girl that could knock you into fits” mean?
Discussed here.
I’ll defer to the natural-born Paddies on this, but I don’t think I’ve ever actually heard this phrase used in Ireland. FWIW.
If someone is “in fits” it means they’re in an emotionally animated state. . “to knock” something in to some state means to force them into that state. So the implication is that Big McGrath knows a girl who could force you (whoever the “you” is in that sentence) into an emotionally animated state.
The phrase doesn’t sound peculiarly Irish, it sounds like something that could be said anywhere in the informal speech of the whole archipelago.
Nor have I. French used some hokey affectations and was writing so long ago it’s possible that if the term had any currency it’s long gone. However, the lexicon of this land is rich enough I’m willing to concede people somewhere may say it. 
With regard to the word fit we have:
“He took fits when he saw the state of his house” - “He got extremely angry/upset…”
“I was in fits when I saw the state of his house” - “I laughed heartily when…”
Are those current elsewhere in the English speaking world?
I’m English, the second example could be said over here but although I understand your first example (and hear it in the voice of an Irish friend) I wouldn’t say it.
He is telling Eileen that he likes her. She is overjoyed at the fact.