Spiritual: Alma's conversion experience was certainly a well known event among the members of the church. It could hardly have passed notice that Alma once persecuted the church, and now lead it. Nevertheless, Alma specifically speaks here of learning these particular things through the Holy Spirit after fasting and prayer. This does not describe his conversion experience. Alma's dramatic conversion may have come after his father's fasting and prayer, but not his own.
Clearly, in spite of the miracle of his conversion, Alma found it important to continue to have revelation from the Spirit, and he approached that continued communication through prayer and fasting. In this particular case, we may presume that he fasted and prayed to know what to say in this particular sermon. In the context of the social background of this sermon, Alma's declaration that he has "knowledge" of certain things may have struck his audience with a dual meaning. Those who were following the gospel would hear it as an affirmation of the truth of the positive aspects of repentance and the ultimate redemption through the Savior. Those who were following what Alma had declared to be the way of Satan might here in this declaration a revelation that their sins had been made known to Alma. They would have known that Alma was not speaking in generalities, but very directly at their actions.