Aside from n-key rollover, which many mechanical keyboards have (essentially, that lets you press all 104 keys at once with no beeps if you're plugged into PS/2), the main reason it because of the feedback that they offer. Rubber-dome and scissor-switch keyboards (the former being found commonly on desktops, with the latter being found most commonly on laptops and on some desktop keyboards) are usually cheaply made. The general feel of the keys typically doesn't lend itself to being good for fast typists, as you generally have to bottom out the keys and don't get very good feedback when they register. There are exceptions (my work keyboard is a rubber dome keyboard and it gives a fairly satisfying click when you press a key), but that is how it goes in most cases.
With mechanical switches, each key uses an actual microswitch, much like an arcade button. Rather than a flexible dome coming down and bridging an electrical contact, each key has its own reliable microswitch inside of the keyboard housing. One major advantage of this is that you can buy a new keyboard and, as long as it uses the same switch as your previous keyboard, the keys will feel about the same when depressed. Because of this fact alone, switch manufacturers are able to introduce things to make it easier to type using the keyboard. The most popular method is adding a small bump on the switch (which is on switches like the Cherry MX Brown), so that you can physically feel when the switch is actuated. This means that you can type without bottoming out the keys, which leads to considerably less stress and noticeably faster typing speeds. Other switches, such as the Cherry MX Blues, feature a two-segment switch design so that in addition to a tactile bump, the keys give a loud, satisfying click when you press them and take less force to push than most rubber dome keyboards.
I'm a fast typist, and simply switching to the Das Keyboard improved my typing speed. Because of how awful the keys on my old keyboard were, I also tend to do better at games -- particularly side-scrollers -- with this keyboard despite it being geared largely for typing.
And just to give you an idea of how clicky a Cherry MX Blue switch sounds, eye maed a vdieo:
http://www.youtube.com/watch?v=U_jvwYaRFVE