For us specifically at Last Leaves, it's just a courtesy thing. We gave the piece first publication and a bit of an audience. Our hope is that publishing a piece helps an author. So, when that author publishes it elsewhere, in a bigger press or maybe in their own chapbook, it's just polite to say "Last Leaves believed in this piece early on".
It helps us get our name out there more, just like we helped the author get their piece out there.