Fetchlands are so good because Brainstorm is so good. If you Brainstorm without a fetchland, you put your two worst cards on top of your library, meaning you're doomed to see those cards the next two times you draw. With a fetchland, you're no longer "Brainstorm-locked."

Deathrite and Tarmogoyfs are considerations too, but they're not played nearly to the extent that Brainstorm is.

Deck-thinning is another oft-cited use, but in practice, the extra land out of your deck is unlikely to give you a statistical edge unless you're shuffling and drawing a lot that turn. It's far better to save your fetches to shuffle away Brainstormed cards.

I think a lot of it has to do with the fact that they can retrieve dual lands. (disclaimer : I'm not an expert)

You could, for example, run a two-color deck with 4 dual lands and 12 fetchlands. You retrieve the dual lands first (if you don't care about Wastelands) and then you get basic lands.

Does that mean that fetch lands are 3 times as popular ? Yeah, kinda, but if you had a choice between using only dual lands or only fetch lands, maybe the decision would favor dual lands in many contexts.

Basically every multicolored legacy deck runs 5-8 fetchlands, some even more. Almost no one runs that many dual lands though; even the greediest decks rarely have 4 of any single land, but every B/U deck is going to have 4 Polluted Delta to get their basics. Additionally, Delta can be played as a 9th fetch in a B/G or U/W deck too, because it still gets islands and duals, but you wont see an Underground Sea in that list.

I believe they are more popular since it offers more flexible.

You can get dual lands, effect lands like wasteland, and regular lands instead of being stuck with any given one of the listed.

You also get the benefit of lowering the odds of drawing a land since you've cleared another one out of your deck.

Finally you get a chance to shuffle (which might matter in some decks).