Fauxbourdon, or falso bordone (Italian), is a way of setting psalm tones from Gregorian chant. Typically the psalm tone formula will be buried in one of the interior voices, and may have been modified to make accommodations for the harmonies constructed around it.
The one heard in this video is set for three mens' voices with the chant formula in the middle voice. Also, as demonstrated here, falso bordone are sung in alternation with the actual chant.
The trick in preparing and executing falso is to determine which psalm tone formula you're using and then start digging through historic collected editions and places like the CPDL to find falsos that match. The even trickier part is that most falsos are just the music; it is left up to the musician to set the texts. In the example heard here, all three voices are moving homophonically, which is fairly easy to deal with. In the more complex (and more beautiful) falsos of Vittoria, for example, the rhythmic motion of the decorating voices sometimes does not line up in a logical manner with the text/notes of the tone carried by the inner voice. There are some understood "rules" about how to set the text, as explained to me by a colleague who had researched it through various sources (including musicologists from the Oberlin Conservatory and Eastman), which include concepts like not setting a particular syllable in any other voice until that syllable has been sung by the voice carrying the original chant melody. Not an easy feat,