Four short lines carry a big, tangled feeling: the speaker is getting rid of letters once written to someone else. The detail that matters most is "in my blood." Blood suggests these weren't ordinary letters. Whether the phrase is literal or figurative, it makes the writing feel painfully personal, as if a piece of the speaker went into every page.
That's why throwing them out feels like two opposite things at once. It is "the most relieving thing" because the speaker is letting go of a weight, perhaps a love or a hope that has been hard to carry. It is also "the most heart shattering thing" because those letters held real feeling, and destroying them means admitting it's over. The poem doesn't pick one reaction over the other. It simply sets them side by side, and the plain, repeated phrasing makes the pain and the freedom feel equally strong. The mood is quiet, raw, and bittersweet.