It's a reference to a collection of Judaeo-Christian poems in Greek known as the "Sibylline oracles", written ca. 2nd to 6th centuries CE. Book 2 in particular contains a narrative of the the end of the world and the horrible things that are going to happen to non-Christians one day; hence the opening lines of the Dies irae, "The day of wrath, that day / will end the world in burning ash / according to David and the Sibyl".

These poems were in turn called "Sibylline" to add to their prestige by associating them with the Sibyls of various ancient pagan oracles (notably Cumae). It's a gesture to affirm the certainty of what's in the poems: "look, even the pagans predicted that this stuff would happen". There were a number of older pre-Christian (and shorter) verse "oracles" attributed to the Sibyls as well, but none of them are thought to be authentic.

Edit. The Dies irae actually echoes Orac. Sib. 2 elsewhere too: at Orac. Sib. 2.339-40 (416-17 in the translation) the "Sibyl" says

αἰαῖ ἐγὼ δειλή, τί γενήσομαι ἤματι τῷδε
ἀνθ' ὧν ἡ δύσφρων γε πονησαμένη περὶ πάντων...

Alas, what shall become of me, miserable one, on that day
in return for all the things I foolishly busied myself with...?

While in the Dies irae we find:

quid sum miser tunc dicturus?...
ingemisco, tamquam reus...

What am I, miserable one, to say then?...
I lament, as a defendant on trial...