The early followers of the Way had a recurring problem with more fundamentalist Jews (called "the Circumcision", including members of both the Pharisees and the Sadducees) treating Hellenists (less traditional Jews who had adopted more Greek culture) and former gentiles as common or second-class citizens. In this situation, the local congregation was dominated by the Circumcision who made sure that their own widows were given preference in the distribution of charity.
When the Hellenists complained, the Twelve agreed that it wasn't right and asked the other disciples to nominate some men from among themselves to manage care of the widows. They nominated Stephen and six others who were confirmed by the Twelve.
The passage (Acts 6:1-7) is arranged chiasticly. On its own, that's not so remarkable; it's a logical arrangement and could have been entirely unconscious on Luke's part. Except for that bit about the priests becoming obedient to the faith. It seems out of place, but is clearly intended to be part of this passage. So what did Luke mean by including it?
The opening point of the chiasm includes the growing number of believers and the resulting conflict between the Jews and Hellenists. The nature of a chiasm is that the final point mirrors the first point or at least corresponds to it in some meaningful way. The final point in this chiasm again includes the growing number of believers, but instead of adding the conflict, it adds the resolution to the conflict: a large number of priests, who were predominantly Sadducees, became obedient to the faith.
The priests in v7 were either the same as the Jews in v1 or else they were instigators of the problem, advising those who had been handling the distribution unfairly before. In either case, they were impressed by the way the disciples managed the situation, and placed themselves under the spiritual authority of Twelve.