"Come on in" has the same meaning as "come in" but is a more folksy way of extending the invitation. It suggests a kind of rural, down-home hospitality that is redolent of (American) TV shows of the '50s, which were ever a myth (although a persistent one) about how friendly people in the hinterland were.
This is an AmE usage, so I wouldn't expect to encounter it in the same way in Britain, though it may be available in some of the many dialects there.
For those who can't get past my suggestion that this is a "folksy" expression, note that I said it is "redolent of (American) TV shows of the '50s," not something like "absolutely 100% a hinterland expression."
Also, the fact that it can be used in Britain proves nothing except that the English language is flexible enough to express an idea in multiple ways. The fact remains, the expression is something heard quite a lot in the US, and carries with it enough regional overtones to be describable in that way.