First, you need to understand that poetry used to play a much bigger role in day-to-day American culture than it does now. Check out this image from the New York Tribune in 1842, which I came across as I was working on an article about how Ralph Waldo Emerson built his reputation in the early portion of his career:
There is one other thing I want you to know as you try to understand why Whitman is so special: Whitman was completely unlike any American poet who came before him (and probably unlike any English-language poet before him). Here is an example of what was considered quality poetry in the mid-1800s:
“A Psalm of Life”
BY HENRY WADSWORTH LONGFELLOW
What The Heart Of The Young Man Said To The Psalmist.
Tell me not, in mournful numbers,
Life is but an empty dream!
For the soul is dead that slumbers,
And things are not what they seem.
Life is real! Life is earnest!
And the grave is not its goal;
Dust thou art, to dust returnest,
Was not spoken of the soul.
Not enjoyment, and not sorrow,
Is our destined end or way;
But to act, that each to-morrow
Find us farther than to-day.
Art is long, and Time is fleeting,
And our hearts, though stout and brave,
Still, like muffled drums, are beating
Funeral marches to the grave.
In the world’s broad field of battle,
In the bivouac of Life,
Be not like dumb, driven cattle!
Be a hero in the strife!
Trust no Future, howe’er pleasant!
Let the dead Past bury its dead!
Act,— act in the living Present!
Heart within, and God o’erhead!
Lives of great men all remind us
We can make our lives sublime,
And, departing, leave behind us
Footprints on the sands of time;
Footprints, that perhaps another,
Sailing o’er life’s solemn main,
A forlorn and shipwrecked brother,
Seeing, shall take heart again.
Let us, then, be up and doing,
With a heart for any fate;
Still achieving, still pursuing,
Learn to labor and to wait.
One chapter in my book Environmental Evasion, and a chapter that I wrote for the book Reconsidering Longfellow make an argument for why poetry like this should be valued, and I think very strongly that there indeed is value in poetry like this, but it is completely unlike anything Whitman ever wrote: it has a simple rhyme, a clear moral message, and relatively little ambiguity in what it is trying to say.
You have already read Whitman—you should be able to see the difference. Whitman is not concerned with rhyme, he is not so moralistic, and all of his poems contain multiplicities of meaning instead of a single purpose as we see in “A Psalm of Life.”
To try to help you understand how people may have regarded Whitman when he appeared on the scene, I want to put the Whitman/Longfellow binary in the context of some more recent artistic shifts.
Before The Beatles, the Buddy Holly ruled the American pop music scene.
After The Beatles rose to prominence, “rock and roll” was confronted with another epochal shift with Jimi Hendrix and Led Zeppelin.
In the 1980s, “hair bands” like Poison controlled the rock scene until Nirvana arrived and effected another shift.
If you wanted to look at something like rap or hip-hop, you could use the Sugar Hill Gang and 2 Live Crew as examples. If you are too young to remember when 2 Live Crew arrived, they sparked probably the biggest censorship discussion that I can remember. The song I’ve hyperlinked to is generally offensive (to the point that some people wanted to label it obscene and have it banned), so don’t be shocked if you watch it. I chose that particular song because it's such an extreme example.