There has always been a certain amount of ambiguity about this lyric, and I'm curious what it might have meant in the context of the contemporary culture or in the context of Lennon and McCartney's collaboration. It seems hard to believe that this band was promoting some sort of underage tryst, but I really have no idea what to think.
Again, this question is completely serious.
Even in 60s pop culture, "You know what I mean" has the risque undertones we associate with the phrase. Paul actually has a quote in the Anthology liner notes about that specific line:
We were learning our skill. John would like some of my lines and not others. He liked most of what I did, but there would sometimes be a cringe line, such as, 'She was just seventeen, she'd never been a beauty queen.' John thought, 'Beauty queen? Ugh.' We were thinking of Butlins so we asked ourselves, what should it be? We came up with, 'You know what I mean.' Which was good, because you don't know what I mean.
They could have meant all kinds of things, and that's the point. It's just as scandalous as you want it to be. But keep in mind that Paul was just 20 years old when he was writing this in 1962, before their big break into superstardom, performing for an even younger Liverpool crowd.
OP: For future reference you don't need a [serious] tag in /r/AskHistorians. Answers to all posts in this subreddit are expected to be serious.
The age of consent in the UK is 16, and has been since 1956, so if we assume it is an innuendo then it isn't about an underage tryst but that the girl is 'just' of legal age.