Why did the children decide to abandon the man after he figured out their names? Why was it necessary for the man to know of their names? (The Man in the Well by Ira Sher)

As "The Man in the Well" is such a complex, ambiguous text with no fixed meaning, one can only speculate in answering your question. I would argue that the man felt he needed to know the children's names because he wanted to maintain a connection with the world outside him, the world above the well. Being trapped down a well would be a pretty traumatic experience for anyone; and in such a scenario, it's important to hold on to whatever reminds you of life above ground. Establishing a connection to the boys through learning their names reminds the man of the common bond of humanity he shares with them.
As for the boys, what seems to upset them more than anything is that their names are blurted out accidentally. But in any case, they're certainly not happy that the man in the well now knows their names. As they don't know his name, this creates a kind of power imbalance between them which makes them rather uncomfortable, to say the least. Abandoning the trapped man to his fate could be construed as the boys' way of restoring the balance.