This isn't like the catacombs of Rome, an archaeological excavation of tombs. Here the bodies were always meant to be seen, and they charge you a small fee for the pleasure. There are signs to remind you to be respectful and not take photographs, but they sell them. It's not clear if this is a religious experience or a cultural one, but it is a tourist attraction.
The first and oldest mummy is a friar: Silvestro da Gubbio, standing in his niche since 1599. (The word "mummy" is from an Arabic word for bitumen, which resembled the blackened resin the ancient Egyptians used as a preservative.) Most of the bodies are from the 19th century. To begin with, they were exclusively friars and priests attached to the monastery. As time went on, the religious men were joined by benefactors and dignitaries and notables.
No one knows exactly what started the mummification; probably by chance it was discovered that a body left in a crypt with a particular atmosphere of coolness and porous limestone would actually dry out rather than rot. Then a system was devised. The newly dead were laid in chambers, called strainers, on terra-cotta slats over drains, where their body fluids could seep away and the corpses slowly desiccate, like prosciutto. After eight months to a year, they'd be washed with vinegar, put back in their best clothes, and either placed in coffins or hung on the walls.
Preserving ancestral bodies is done in any number of places, but they're rarely displayed like this. Sicily has so many cultures, so many people came here with their practices and beliefs and were assimilated, that little bits now and again rise to the surface, their origins long forgotten. It has been suggested that perhaps the practice is the residual echo of a much older, pre-Christian rite—belief in the shamanistic power of corpses. Not every corpse would have dried out; some must have rotted, and so the preservation of others might have been an intimation of God's will, a divine hand keeping certain individuals as they were as a mark of a particular worldly goodness. As saints' relics are used to aid prayer and belief, maybe these bodies were thought to have been preserved by God to reinforce faith. Or perhaps the catacombs were made as a great vanitas, a memento mori, an illustration of the passing of all worldly ambition and the inevitability of death and the vanity and foolishness of storing up wealth on Earth.