Whitman doesn't want us to think that his thoughts are especially original. If he were truly saying something that no one had thought of before, it would undermine his entire point about the connections between people.
If we don't take his words as our own, we won't be able to get anything out of them.
Not only his thoughts, but also the grass he walks on and the air he breathes are ours.
Remember, though, that "grass" also refers to the "leaves" or pages of the poem. (Title Alert! Whitman published "Song of Myself" in his book Leaves of Grass. More on that in "What's Up With the Title?")