There are quiet mechanical keyboards. They will cost you $200+ because they use Topre switches. Cherry MX-based boards that are advertised as "quiet" or "silent" almost exclusively use Cherry MX Brown switches. The switches are silent. If you type on a mechanical keyboard as it's intended, it will be silent. the problem is that 99.9% of people do not type this way and end up bottoming-out the keys.
Bottoming a key just means that you push the key until it stops moving instead of pushing it only until it actuates. What stops it from moving is the back of the keycap hitting the plate/PCB. This results in a moderate to loud 'clack'. There are only two ways to keep this from occurring. Either stop bottoming-out the keys, or get dampeners as thomas lansky has already pointed out. The problem with the o-rings/other dampeners is that they make the keys feel mushy. It'll always be a mechanical keyboard and the key repeatability and most other benefits will always be there, but you'll essentially lose the solid feel that you get with each undampened keypress. In addition to that, you also lose a certain amount of key travel. Since it prevents you from bottoming out, it shortens the distance that you can press the key before the key hits the dampener. To me, this just feels awkward.
I personally cannot stand using a mechanical with dampeners, but I know of several people who prefer to use them.
http://elitekeyboards.com/products.php?sub=access,slpads
These are the least offensive-feeling dampeners I've used. I would recommend the firm ones as they don't make your keys feel as mushy as the soft ones.
This video should give you a pretty good idea of the difference between dampened and undampened keys.
https://www.youtube.com/watch?v=lFkl1Vet1eU