There are a couple of important flavors of apostrophe in this poem: apostrophes to the landscape, and apostrophes to the people of the future. Addressing the river and his readers with equal ease, the speaker suggests that people and the world are all somehow together, sharing one big, eternal consciousness.
The entire first section of the poem is an apostrophe and sets the tone for all that follows. First, take a look at the speaker's address to the world around him:
Flood-tide below me! I see you face to face!
Clouds of the west—sun there half an hour high—I see you also face to face.
To this speaker, the personified "flood-tide" of the river below, and the "clouds" and "sun" above, are creatures he can address "face to face," people he can greet with exuberant affection. The whole world, in his eyes, is alive. Calling out to the river and the sky this way, the speaker seems at once worshipful and informal: he's meeting the world itself as a friend and an equal, but also as something astonishing, something to be addressed with a degree of formal respect.
When, a moment later, he addresses plain old people, his tone gets a little bit gentler:
Crowds of men and women attired in the usual costumes, how curious you are to me!
On the ferry-boats the hundreds and hundreds that cross, returning home, are more curious to me than you suppose,
And you that shall cross from shore to shore years hence are more to me, and more in my meditations, than you might suppose.
From greeting the whole world "face to face," the speaker turns now to noticing how "curious" the people around him seem: in other words, they seem almost strange, inspiring his curiosity and fascination. That feels like a more everyday feeling than the sense that one is greeting a river.
But things get more mystical when the speaker starts addressing people who haven't even been born yet—a moment that might give modern-day readers, approaching this poem more than 150 years after it was written, a little jolt.
In this speaker's eyes, there seems to really be no difference between speaking to the world, the people around him, and all the people who will come after him. As he puts it in line 91: "Who knows, for all the distance, but I am as good as looking at you now, for all you cannot see me?"
Apostrophe is thus at the very heart of the poem's philosophy. To this speaker, everyone that has lived, lives, or will live is within his reach, and so is the whole world: he can speak directly to everybody and everything.