As I understand it from my years of lurking, generally what is meant by "flat response" is that the headphones impart the sound as close to the source as possible. Many headphones both amplify/emphasize and/or mute/de-emphasize parts of the sound (such as the bass, the vocals, etc.), and lots of people don't like this.
An anecdotal experience of this for me is my other set of "full-size" headphones - the Sol Republic Tracks HD. When I got those headphones from my campus store, I thought they sounded really awesome; lots of bass, they could play really loud, and they looked pretty cool. But as I kept reading about sound quality online, I found myself comparing the impressions people were giving about other sets of headphones to what I was hearing from the Tracks. I started to notice that there seemed to be a definite difference in sound quality between those impressions and my experiences.
A couple of things pushed me further towards getting better headphones; for one, I ran across a sound demo video from Zeos on YouTube that blew my mind, and for another, I tried out different sets of headphones at tech stores when I was out and about. I tried out Beats, Sennheisers, Bose's, quite a few. And I found that I was indeed missing out when it came to my Tracks HD. For the Tracks, the bass is really emphasized, and the general sound is kind of muddy. Instruments can be pushed to the background, while vocals can be pretty unclear. And the soundstage is quite narrow; there's not a lot of depth to the sound.
I absolutely love music, so discovering that the music I was hearing was being delivered in a "biased" way (a.k.a emphasized or de-emphasized too much; this is referred to as "coloring") made me want to hear the music as the artist intended it. When I listen to the Deftones, I want to hear the music how Chino, Abe, Frank and the others wanted me to hear it. When I listen to Tesseract, I want to hear the little atmospheric touches that are put in place to draw the listener in. The examples go on and on.
Of course, there can be "downsides" to having "flat"ter headphones. As the phrase commonly goes around here: "Garbage in, garbage out." If you are listening to badly produced/mastered songs, you'll probably notice it. A lot of the music you previously liked might very well become unlistenable for you depending on your new headphones. There are ways around that, but in general, those are the risks you can run.
But for me, it's pretty much worth it, because I can enjoy the music as it was meant to be enjoyed (usually). Without falling too much into the trap of comparing sound to food, it's like eating food that's spiced too heavily, and food that's spiced just right, or at least closer to just right. Don't get me wrong: I'm Caribbean, sometimes we like food with a LOT of spice to it! But that, like sound, is based on our preferences; some Caribbeans don't like spicy food at all. Similarly, some people like the sound they get from a particular set of headphones (like Beats).
For many people (on Head-Fi and elsewhere), the ATH-M50x are too bassy, and the sound can be somewhat muddled. I have found this to be the case for me as well when I tested a pair of M50x's vs a pair of M40x's at Sam Ash. To me, the M40x's just sounded clearer and more "true" to the sound than the M50x's. However, from what I've read, the M40x's aren't truly "flat" either, they're just closer to "flat" than the M50x's are.
But plenty of people (here and elsewhere) like the sound of the M50x's and say that the M40x's don't have enough bass for them, and to that, I say "More power to you; find the right sound for you".
Some people walk the middle ground; they like the sound of the M50x's, but agree that there's too much bass, and so they perform mods or switch out the pads to change the sound, reduce the bass a touch, and clarify the sound. That's great too.
In the end though, sound preference is very subjective, so you have to pick what's right for you. That doesn't mean don't get out of your comfort zone and try new things, because hey, you might like it; it just means don't let someone tell you you're a neophyte for not being able to hear what they're hearing or liking what they're liking.
So if you like your M50x's and don't care about them being "flat", I support you dude
Sorry for the rant, I can get too into writing sometimes