How much time do you have?
First, it is important to understand that it is not just Mets fans that dislike the Yankees; it’s pretty much fans of every team, except maybe Devil Rays fans, because they don’t actually exist.
The reasons are myriad; here are a few:
(1) The Yankees always win. For this reason, the Yankees winning is boring. Predictable outcomes are seldom exciting outcomes. I want the Yankees to miss the playoffs because I want to see a different playoff lineup for once in my adult life, one that doesn’t involve pinstripes.
(2) The Yankees spend a lot of money, more than anyone else. It is true that the Red Sox and Mets (and a few other teams) have done this as well, but they have done it in an effort to keep pace with the Yankees, who otherwise wouldn’t even be on the same continent. Steinbrenner can afford to throw good money after bad to patch problems that other teams have to live with. It gets to be a bit much after a while. Look at the team this year; they’re already paying kajillions of dollars and most teams would either have to live with the roster they have or make a clever trade. The Yankees don’t have to make a clever trade. They’ll just hire Roger Clemens to ride in on his bitter and vaguely psychotic horse, and acquire, I don’t know, Todd Helton or somesuch for two grade D prospects because they can afford to pay his salary and Colorado can’t.
Now in fairness, we don’t complain this much about the Red Sox, and they spend a lot, too. But it’s kind of like when the kid down the street picks on the smaller kids for years, and then one of them finally gets pissed and works out and gets strong enough to challenge the bully. Sure, maybe the newly-strong kid will now turn around and kick your ass, but at least the bully’s getting a black eye for once.
(3) We have to root against someone. When your team is going into the tank, you need a reason to watch the games. Rooting against the Yankees has made me, in essence, a fan of 30 major league teams, and given me a rooting interest in players I’d otherwise never have noticed (Dave Roberts, say, or Luis Gonzalez, two of my sports heroes).
(4) Yankees fans do not win well, as a group. They attribute a ridiculous amount of their team’s success to “mystique” and “Yankee pride” as opposed to “All-Stars at every position” and “highest payroll in the Seventh Circle.” This makes fans of other teams, who (we are to understand) are not quite so blessed, irritable.
I admit to having a special hate-on for this particular Yankee team, though. I really have no problem with Derek Jeter (I know many do, but I think he’s an exceptional player if slightly over-rated by the typical Yankee fan), think Jorge Posada is a big asset, and would take A-Rod on my own team any day of the week, even if he only “chokes” his way into 40 home runs or so. But there are three players - Giambi, Mussina, and Damon - that I can never root for, as long as I live. Here’s why:
I mean, there’s a certain weird ethic to sports. When I used to play pickup basketball, sometimes teams would get picked such that one team beat the other pretty soundly. Afterward, someone from the winning team would invariably say, a little patronizingly, “switch teams?” But you didn’t do that, if you were on the losing team; you went again, and if you lost again so be it, but if you couldn’t beat them you certainly weren’t going to join them.
That’s what Jason Giambi did. He joined them. He was the best player on a good team that kept losing to the Yankees. The kind of player I’d respect would have said, Hell, no I’m not going to the Yankees. I’m going to get my ring by beating the best. And he would have stayed in Oakland, and tried to get past the Yankees. Instead, he meekly called out, “switch teams?” Went to the Yankees in a craven and obvious effort to get an easy championship.
Ditto Mike Mussina.
As for Damon: leaving Boston for the Yankees is like dumping your girlfriend for her bitchy sister who has bullied your girlfriend all their lives; you just don’t do that.
But I’ll tell you what? Ditch those three, and I’ll root for the Yankees to win the wild card. It’s the least I can do. 
