Amulek began by saying, "I am a man of no small reputation." In today’s terms, he may have been saying, "I am credible. This is a hard story to believe, but here’s what happened." It is important to establish credibility when bearing testimony.
He stated that he was descended from Aminadi, who saw the finger of the Lord writing on the temple wall. He seems to be saying, "I come from a heritage of people among whom these kinds of manifestations are taken seriously." In addition, the knowledge of this detail in his ancestor’s experience may well have predisposed Amulek toward recognizing Alma as one bringing the word of God. Potentially, this story may also have been part of Amulek’s reputation among his peers. In a culture like his, the fact that Amulek’s ancestors had received these kinds of manifestations would have been very prestigious. Amulek may also have been saying to his audience, "You accepted that incident which happened long ago. Well, let me tell you what happened to me just the other day."
And notice that Amulek gives a great deal of detail about his own conversion. As people write about their conversions, it is important to have detail there to help others share in the experience. Amulek spoke as someone who knew the truth but who had rebelled against God. As someone who had formerly hardened his heart, he still responded to Alma. This is how he hoped things would also happen for to the people of Ammonihah. When trying to talk to and teach someone, relating to them is crucial. Real communication requires bridges to be built.
And, by the way, Amulek also traces his lineage back to Nephi, Lehi, Manesseh, and Joseph, who was sold into Egypt (10:3). This is the only place in the Book of Mormon where we learn that Lehi belonged to the tribe of Manesseh. The mention of "Joseph who was sold into Egypt" also provides a potential connection with the annual season of Passover.