They are repeating the antiphon (from the Requiem Mass) after every verse of the Miserere, as though it was a Responsorial Psalm. We do it sometimes (in the OF), usually at Communion, to prolong the music when the action is long.

Since the link above doesn't embed in the forum page, here's another version of it:
https://www.youtube.com/watch?v=-EXkKT4XmFk

I'm a new poster, who's part of the same choir Andrew Malton sings in. I'm very interested in the technique demonstrated in this video. There is clearly congregational singing happening. The antiphon is from the Office of the dead so this is not an Ash Wednesday ceremony which has the same psalm but a different antiphon and therefore would have a different psalm tone for the psalm verses. I point out that the version of Ps 23 posted with this on Youtube is also from the Office of the dead. I can imagine a circumstance such as John Paul II's death when crowds of people filed past when a prolonged office with congregational participation would have been the ideal thing to do, but otherwise I'm stumped.
Doing this maximizes the meditative character for the congregation and alleviates the need for participation aids. I would think it is almost a Taize-like experience albeit without instrumental accompaniment, harmony or rhythm. Has anybody ever taken a similar approach in doing a vesper service or any other office liturgy (compline).