Two people, the speaker and someone they love, see themselves as broken, and the poem turns that damage into a reason to be together. The images are all of ruin: ashes blown across a desert, a shattered clay pot, a rock pounded by waves. Each one suggests something that has been worn down by forces it couldn't control.
But the poem doesn't stay in despair. Near the end, the ashes stop scattering and settle into a pile, and the pot pieces become "one from whence they came." The broken things find rest and begin to come back together, which fits the title, since puzzle pieces only make sense when they fit one another.
The speaker's mood is tender and quietly certain. Repeating "We are both broken souls" feels like a confession, while the closing line, "Yes, i know that's why we are perfect for each other," sounds settled and sure. The poem's comfort is that being broken is shared, and so it is no longer lonely.