In Night, how old was Elie when he got to know Moishe?
This is a good question. Right in the beginning of the book, Elie states that he met Moishe when he was almost thirteen. So, Elie was twelve. Here is what the text says:
I met him in 1941. I was almost thirteen and deeply observant. By day I studied Talmud and by night I would run to the synagogue to weep over the destruction of the Temple.
The more important question is why he met Mioshe.
Elie was a spiritually keen and sensitive boy. He had a desire to study the Kabbalah. When Elie asked his father for a teacher, his father said that there was no one who could teach him. This did not deter young Elie. He went about to find his own teacher, and this is how he met Moishe. In Elie's own words, he writes: "I succeeded on my own in finding a master for myself in the person of Moishe the Beadle."
As their relationship grew, Elie learned a lot, but more importantly, one day Moishe was gone. He claimed to have seen what the Nazis were doing to Jews elsewhere. No one really believed him or did not wish to believe him. But Elie was introduced to the horrors of the Holocaust through Moishe.