This poem is I think an earnest, unostentatious declaration of Emily Dickinson's outlook on life and possibly also a sort of mission statement for both her life and her poetry.
Life can get rather complicated and we tend to spend a lot of our time and energy worrying about things which, relatively, really aren't that important. We worry about how much money we might make, what clothes we wear, which people we should impress, and so forth. For Dickinson, the point of life, at least the point of her life, is simply to make somebody else's even a little bit better, or easier.
She wants only to "stop one heart from breaking," or "cool one pain." This may sound rather unambitious, but it is nonetheless a laudable aim. If everybody had this as their mission statement for life, the world would very likely be a much kinder, much safer place.
Dickinson may have had her mother in mind when she wrote this poem. Her mother was left paralyzed by a stroke for the last seven years of her life and was cared for by her daughters, including Emily. Emily later wrote about her mother, in a letter to a friend:
“When we were Children and she journeyed, she always brought us something. Now, would she bring us but herself, what an only Gift.”
This quotation captures the same selfless, altruistic sentiment that is expressed in the poem. Dickinson, it seems, considered it an honor rather than a duty to be able to "cool one pain."