How was the Corleone family being "chased out of New York" as Moe Greene put it? Can someone clarify this?

Moe Greene was referring to the war that the Corleones appeared to be losing. I say "appeared" for a reason. In the novel (and in the movie too, just the novel makes it more clear), the Corleones made it seem like they were weak. Michael created that facade. In doing so, he caught his enemies off guard. He made Barzini believe he was now the Don of all New York. Michael looked weak, young, and inadequate as boss - all for a reason. One of the most memorable quotes from the novel is (Im paraphrasing a bit): There is no greater position to be in than to have your enemies overestimate your faults and your friends understimate your values.

Also with Michael moving his operations to Nevada it could look like he was running away and establishing a different territory for the reasons Sopranorleone mentioned.

Moe told Michael: "I talk to Barzini." Shortly thereafter, Tessio complains that Barzini is opening up so many operations in his territory that "Pretty soon I won't have a place to hang my hat." So it's logical to conclude that Barzini was telling Moe that the Corleones were being run out of NY.

And who better than Fredo to give that misimpression.