I am a descendent of Romantics; I accept this.
How to say that it happens? Usually an image or phrase that words coalesce around. You start writing, the words build on top of each other. You come to the stopping place and look at it all, briefly in love. Themes and connections that seem like they were made consciously appear to you. You're sure there are others that others could find.
And then there's the fiddling, which is sometimes quite short, sometimes extensive. Sometimes 75% gets excised, sometimes only a couple of words switched out. If there's an opportunity for assonance or rhyme, I usually take it. The page gets quite messy, which is exhilarating: a foul copy. You copy it into a fair copy, which becomes another foul copy. Eventually it goes into the computer, but computers aren't as good at being messy.
Some poems are journal entries, more or less chopped up prose, and those have their purpose although I would not send them to be published. The poems that require and reward more work go beyond the meanings of the words.
Because poetry is not just the meanings of the words, which makes it hard to read unless you are prepared to play, be befuddled, be heartbroken. Because words have secret and official histories, because they sound certain ways, look certain ways, because they react chemically with each other when placed in proximity, because they ooze chartreuse guts when broken out of idioms. And sometimes that takes some patience to understand, or grace to let understanding alone.
That's what I'd say poetry is. Quite effortless and quite a lot of work at once. And the person who writes it? They always say that person must be attentive, which is true. Attentive to how anything that enters their experience could become a poem if they let it, if they're willing.