Translation from the Latin: "They gave up all to serve the republic."
First let's figure out what an epigraph is and what its purpose is. It's a short quotation at the beginning of the poem intended to introduce the poem. In this case, it's the last word between the title and actual poem. It's meant to preside over the entire poem, much like the title, but often in somebody else's words.
Here, Lowell very resourcefully takes the Latin phrase that's engraved in the actual Civil War memorial to introduce a poem on the very subject.
Given the title, we can assume the epigraph is used to introduce a poem that will be, in some way, a dedication for the Union soldiers that died in the Civil War.
Quick history reminder of the Civil War: the Union soldiers were from the north, and wanted, among other things, for all of the states (northern and southern) to remain part of the Republic. The Confederates, the opposition from the south, wanted to break away from the Republic.
The other huge difference was that the Union supported Lincoln's decision to abolish slavery and, by and large, the Confederates did not.