I've gradually realized that when I incorporate experience into an essay I almost always improve on a first draft. In a sense, the experience brings the essence. That's not to say I can't write well without relating an experience, but it's almost always better than not, like a team preferring to play in front of its home crowd for a big game; it's a subtle but occasionally decisive advantage. If I reach a point in my work where two roads diverge at the crossroads of experience, I know what will make all the difference.
Why does a reference to experience seem to correlate with better writing? The temptation is to suggest that the experience itself informs the writing, that without experience I wouldn't know what I was writing about, but although this is indisputable I think it's a little too easy to be the best answer; you can write about a walk strictly in common terms, describing the who and the what, the when and the where, but if this were true the AP wire would carry the most compelling writing on the planet.
I think part of the answer is in the definition of a story. At its root, a story is about a decision, and in one sense an experience is everything associated with that decision. This is why the best business writing is told in the first-person; a business is defined by decisions, so compelling business writing must be about the ensuing experiences. It's still possible to write well about business without being involved in the decisions but these accounts are like the stories people tell about the things that happened to them, or in their vicinity; the entire account is underscored by the passive voice, which of course is the object of such stories.
But I'm still left with the question of why an experience unrelated to a decision improves my writing. The best answer I have at the moment is in the root of another word, essay, which in a sense means 'to try'. It suggests I must be trying something each time I start an essay but I often find it lacking until I inject a bit of experience. This makes a lot of sense to me because until I bring myself into the picture, I'm not really trying; even if I think through an idea on my own, it's likely someone else who can lay claim to the initial attempt. They say Newton invented gravity, but many apples fell before one impacted his thinking; scores sing the same song about Trader Joe-san, but only I can tell you why it makes me a background object.