We understand, then, do we not? What I promised without mentioning it, have you not accepted? What the study could not teach—what the preaching could not accomplish is accomplished, is it not? What the push of reading could not start is started by me personally, is it not?
Without even knowing it, we've been negotiating an agreement with our speaker. As they say in the mafia, we've reached "an understanding." Hopefully this doesn't mean that we now owe him our life savings.
Whitman is talking in vague and mystical terms about the power of art and poetry to produce a change in people without them realizing it. But most artists don't make an announcement about it like our speaker does, "See, you're changed now, aren't you!"
Again he takes a dig at school and religion by saying that his poem has fulfilled the goals that teaching and preaching could not. Maybe this stance has something to do with the fact that Whitman was largely self-taught as a poet, an example of what the American philosopher Ralph Waldo Emerson called "self-reliance."