I understand a team moving away from an area where there is little market for a team (particularly if there is already a team, in New York's case, the Yankess) and to an area where there is a bigger market.
Yet thr fact that the Mets were created after the Dodgers left would seek to indicate there was room for a second New York team anyway.
In fact, it wasn't just the Dodgers that moved. The very same year, the New York Giants moved to San Francisco. Major League Baseball wanted to have teams on the west coast and could finally do it now that teams traveled by plane, not bus. But logistics demanded that at least two teams be out there, so that visiting teams could have an actual road trip instead of just 3 games and gone. In addition, both of these teams were in search of new stadiums, and therefore amenable to the idea of leaving New York. It even helps that they had a rivalry that wouldn't be disrupted if they both moved at the same time.
So that's why both teams moved when they did, and it probably didn't hurt them, fan base wise. Yes, they had very large fan bases, even splitting New York into 3, but now each team had a major metro area of its own (because this was before the As and the Angels were in California). But you're right, that did open up space for another New York team, and the founding of the Mets did just that.
Interestingly (to me, anyway), each New York team has represented a different borough. The Yankees are in the Bronx, and the Mets in Queens (except for their first few years in the Giants' old stadium, the Polo Grounds). The Giants were in Manhattan, and the Dodgers were (obviously) in Brooklyn. Staten Island is left out in the cold again.