Just as you feel when you look on the river and sky, so I felt, Just as any of you is one of a living crowd, I was one of a crowd, Just as you are refreshed by the gladness of the river, and the bright flow, I was refreshed, Just as you stand and lean on the rail, yet hurry with the swift current, I stood, yet was hurried, Just as you look on the numberless masts of ships, and the thick-stemmed pipes of steamboats, I looked.
At this point, it sounds like he's talking directly to us, the readers. And not just us as fictional readers; it's like he's reaching off the page to grab us by the collar. "I'm just like you!" he says, shaking us back and forth.
All those spiritual experiences that you think nobody else understands – he's been through it. The exhilaration of being in a crowd, the thought of being carried away by rushing water, the amazement at a port full of boats: he knows the feeling.
Whitman is battling against the experience of "alienation," which is a fancy word to describe the way people feel disconnected and isolated from one another, even when they are going through the same thing.
Whitman thinks that alienation is a bad thing. A very bad thing. Did we mention that this poem was first published near the start of the Civil War?