As the original article points out, these weren't specially-prepared corpses like ancient Egyptian mummies, but "spontaneous" mummies which were preserved due to local conditions that prevented many of the processes of decomposition. You can read more about those processes in The Scientific Study of Mummies by Arthur Aufderheide. In particular, the article cites "A combination of low temperatures, ventilation and low humidity" which would dry out the corpses and inhibit normal bacterial breakdown of tissues.

How did the Catholic Church come to be involved in mummification?

Various Christian sects practiced embalming and deliberate desiccation to create relics (Aufderheide 66). More generally, Catholics and Eastern Orthodox Christians have a tradition of "incorruptibility" with the bodies of saints, which is essentially spontaneous mummification (although attributed to be miraculous). You can read about it in detail in Christian Mummification: An Interpretive History of the Preservation of Saints, Martyrs and Others by Ken Jeremiah.

Were monks often mummified, or was it an honor or something?

In this particular case, it doesn't appear to be deliberate or in any way systematic. Various Christian monastic sects did have to deal with the internment or disposal of their members when they died, and had various means to do so. According to Jeremiah mummification was "an accepted and relatively common practice of Christian monks" (50).