"Through faculties given to me by the Archbishop, they were dispensed from banns."
"By virtue of a faculty given to me by the Archbishop, they were dispensed from banns."
(And, for the honours students, "ARCHREPISCOPO" is misspelt in both transcriptions. It should be "ARCHIEPISCOPO".)
It means that not only did Stephen and Mary (and, previously, Stephen and Anne) get a dispensation from the usual obligation to have banns of marriage published, but they got the dispensation not from the Archbishop, but from the priest who married them, he having been authorised by the Archbishop to give dispensations on the Archbishop's behalf.