Types of EDH mana base
Fetchlands + duals is a formula used by almost all the top decks in every format that allows them. The main advantage is monumental consistency.

Verdant Catacombs, for instance, can fetch any of the following original dual lands:


Badlands (Swamp / Mountain)
Bayou (Swamp / Forest)
Savannah (Forest / Plains)
Scrubland (Plains / Swamp)
Taiga (Mountain / Forest)
Tropical Island (Forest / Island)
Underground Sea (Island / Swamp)


That translates into easy access into any color you need, significantly superior to even what you'd get running a full set of Shards of Alara tri-lands.

In addition,


The duals you fetch can produce two colors, so you'll have good color depth among the lands you have in play (color depth is an advantage of multi-color non-basics in general).
Fetches provide a way to shuffle away the top of your library, commonly employed with cards like Brainstorm and Sensei&#39;s Divining Top: put the cards you want on top of your library and the cards you don't underneath them, draw the cards you want, then crack your fetchland to shuffle the chaff away.
Many multi-color lands come into play tapped at least some of the time. Both old-school duals and shocklands can come into play untapped, so there's no loss of momentum compared to playing basic lands.
Since cracking a fetchland actually removes another land from your library, it thins your deck, reducing your likelihood to draw lands in the late game when you'd rather have spells. This effect is rather marginal, but it's a tiny little bonus.
Fetches basically double your landfall triggers.
Unlike other non-basics, these lands make it very easy to use domain-like abilities. This is basically why "Zoo" even exists as a deck type in Modern and Legacy. In Commander you'll see this used to support some powerful cards like Collective Restraint.
Fetchlands give you an easy way to put lands in your graveyard, which powers up Deathrite Shaman, Tarmogoyf, Life from the Loam combos, threshold cards, and other stuff.




Why shouldn't you play fetches, duals, and shocks, besides the cost of owning the cards?

Strategically speaking,


Fetch abilities can be countered by a few rare effects. In Legacy, the spell for this is Stifle. In Commander, you could fall prey to repeatable effects like Azorius Guildmage or Voidmage Husher.
Fetches are vulnerable to search hosers like Aven Mindcensor and Shadow of Doubt. Commander has a very powerful hoser: Stranglehold.
A heavily non-basic mana base is vulnerable to non-basic-land hate like Wasteland, Back to Basics, Blood Moon, Price of Progress, and Ruination. Experienced players can easily mitigate many of these effects by using their fetchlands to get basic lands, though; ultimately you still have better protection against them than you would with a multicolor deck that doesn't use fetches and duals.
Fetches and shocks drain your life. A couple life here and there seems inconsequential, but it can add up pretty quickly.


Also, practically speaking, shuffling your deck all the time is annoying. I can't even shuffle 100 sleeved cards without splitting the deck in half. Although you can at least reduce some of the inconvenience by cracking fetches on an opponent's turn (similar to proper Divining Top etiquette).