A simple answer to how it is a romantic poem is to state the poem is written by Samuel Taylor Coleridge. He, along with William Wordsworth, is one of the most famous Romantic authors. He and Wordsworth practically started the movement. Of course that doesn't mean everything Coleridge ever wrote is a prime example of Romantic literature, but "Kubla Khan" definitely is.
One thing to associate with Romanticism is a focus on nature. And it isn't just a simple fascination with nature. It's a reverence for nature. It extends practically to the point where nature is a mystical being that can grant clarity to those who are lucky enough to commune with it/him/her. "Kubla Khan" features the nature motif throughout the entire poem.
"A stately pleasure-dome decree:
Where Alph, the sacred river, ran
Through caverns measureless to man
Down to a sunless sea."
Notice the focus on nature, specifically water elements. Then, a few lines later, Coleridge switches to dry land descriptions.
"And there were gardens bright with sinuous rills,
Where blossomed many an incense-bearing tree;
And here were forests ancient as the hills,
Enfolding sunny spots of greenery."
I especially like "enfolding sunny spots of greenery." It sounds so human. Enfolding is wrapping or surrounding or enclosing, etc. I have little kids, and when they get hurt, the best form of comfort is to "enfold" them with a big hug or blanket or something. I don't know why it works; it just does. Coleridge's description of the forest and its closeness is very warm and comforting, like being enfolded in a parent's security. The second stanza is along the same lines as the first, but this time the speaker talks about a large canyon that the river flows through.
"And mid these dancing rocks at once and ever
It flung up momently the sacred river.
Five miles meandering with a mazy motion
Through wood and dale the sacred river ran,
Then reached the caverns measureless to man,
And sank in tumult to a lifeless ocean..."
Again, the reader can see how Coleridge is making nature into almost a physical being. He has rocks dancing and the river flinging. It's all so big and amazing to him that it is completely "measureless." It must be a truly beautiful place. That reverent focus on nature is why "Kubla Khan" is a good example of Romantic poetry.