When I was one-and-twenty (1, 9)
Here, the speaker is playing off the basic idea that young = stupid.
I heard a wise man say, (2)
Calling someone else wise means that, well, our speaker is comparatively less-than wise. That's another name for foolishness, friends.
And I am two-and-twenty, And oh, 'tis true, 'tis true. (15-16)
Here's the moral of the story: young man grows older. In so doing, he gets over his youthful folly. He's growing up, see? We can tell that he gets the point – if only because he starts to repeat himself to emphasize his point!