Why would anyone want to confess to a monk or a nun rather than a member of the Clergy?
Because, believe it or not, there are people with incredible spiritual discernment who are not ordained.
For example: many times the women at a Women's Monastery will confess to the Abbess; part of this is due to the fundamental relationship between monk and master (which is also seen in many monasteries where the monks confess to their Abbot); part of this is due to the incredible spiritual maturity that is often exhibited by the Abbesses. The abbess will hear the confessions and provide spiritual support and direction, and when the priest (who serves the monastery) comes by, he will ask the abbess for the names of people that he needs to read the prayers over. He will then see each of them and complete the sacrament.
With the Male monks, sometimes the Abbot is a priest, and so they can go to him for guidance, confession, and he will read the prayers. But often times the Abbot is not a priest, yet has the gift of discernment, and will hear confessions in a similar manner to the Abbesses.
This doesn't even include the Elders who are not Abbots or Abbesses who are gifted and not ordained, who have still been blessed to hear confessions.
The common thread amongst all these stories: permission from a clergyman has been granted beforehand! In the case of an Abbot or Abbess, it is the Bishop who enthroned them that grants such permission. In the case of elders, it is their abbot or abbess, who has the blessing to act as a bishop in many ways within the monastery. In every case, permission from above comes before a non-clergyman begins to hear confessions.