I've heard that it helps clear the beer, but how does it do that, and how is it used? Further, are there any tradeoffs associated with using it?
Irish moss is added to the boil for the last 10-15 minutes. It acts as a clarifying agent by attracting proteins and other solids in the wort, causing them to form larger particles that are more easily removed and/or flocculate out. Consequently, it reduces chill haze in the beer later on.
Although it doesn't exactly smell pleasant by itself, it doesn't add any flavors or aromas to the beer, especially because so little of it is needed to take advantage of its clarifying effects.