Why did Aunt Nicey believe the unusual circumstances would be important?

From the text:

Everybody thought he was going to die-everybody except Aunt Nicey, who had delivered him. She said he would live because he was born in a caul, and cauls were made from Jesus' nightgown.

Back then, in rural communities, and in the African American community, they had traditions and superstitions that were passed down through the generations. One of them being a baby was born with the amniotic sac still intact was special. This is most likely based on the rarity of this. As most babies are born without the membrane. It usually ruptures either before or during labor. It's the "water breaking ".