It wasn't a quick process, it took a while. The Church was not 'extinguished' with the loss of the Apostles, it 'dwindled' away, like a fire burning out without anyone left to tend it.
James E. Talmage does an excellent job of describing the process in his book The Great Apostasy.
It began with the deaths of the apostles. They held the keys to administer in the government of the Church, when they died, those higher priesthood keys were lost, but this did not mark the end of priesthood authority on Earth. The Churches that had been established by the apostles were still led by ordained priesthood holders, such as Linus, who was the bishop of the Church in Rome. These priesthood leaders continued to teach and preach, but without continued direction by revelation, the doctrines of men began to enter the Church. The addition of ceremony was one of the factors that contributed to leading the Church into Apostasy, simple worship services were (and in many cases still are) unusual to many people. Worshiping God was popularly associated with elaborately decorated halls, expensive robes, symbols, and ceremonies. Without inspired direction from the head of the church, these things crept into Christian worship services.
Not having the authority to ordain new Priesthood holders was another significant factor that led to the Apostasy, as higher priesthood authorities died off, the church lost the ability to ordain new bishops and priests. This is why I think Bishops hold such high status in the Catholic Church today, as the higher authorities died, the Bishops became one of the higher positions of authority that remained in the Church.
This mingling of the doctrines of men with the Church continued to the point where there was much contention over the identity of the Church and it's doctrine, this is what influenced the creeds. If there is to be a marker for when the Church truly ceased to exist, and the Great Apostasy was complete, then I suppose you could consider that time to be the creation of the creeds, when the very identity of God was lost.
There was a lot more going on, but the short answer is; The priesthood was slowly lost as leaders died, unable to pass along all of their priesthood keys.