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This is a list of a competitive deck that uses fetchlands in a mono-colored deck: Goblin 8 Whackers.
I don't see the reason someone might build something like this, aside from a slight bluffing that you might be using another color aside from your main one.
Is there an advantage to losing one life and fetching a mountain? Maybe to shuffle your deck?
One reason is that it thins your deck.
It is generally known that smaller decks work better than larger decks, which is why almost everyone uses the smallest legal size of 60 in constructed.
Cards that draw you cards when you play them (cantrips such as Gitaxian Probe) turn your 60 card deck into an effectively 59 card deck. Fetchlands do the same thing. After using a fetchland, you have removed a card from your library.
Note that unlike "draw a card", searching for a land specifically removes a land from your library. This means that you are more likely to draw a non-land card after you have used a Fetchland. So it effectively changes the land to non-land ratio of your deck slightly.
So the deck in the example has 19 lands. If they were all Mountains, then after you play your first mountain (pretending you had 0 cards in hand or anywhere else), you would have a 18 out of 59 chance that the top card of your library is another Mountain. If instead you play a Bloodstained Mire first, and use it, then after that you have a 17 out of 58 chance that the top card of your deck is a Mountain. More chance of drawing non-lands instead.
In that particular deck, the loss of life is not very relevant; that type of deck is hoping to win the game before the opponent can deal 16 or so damage to you. So having 20 life instead of 16 isn't going to save you. In a different type of deck, they might use Evolving Wilds or Terramorphic Expanse instead. They have the same deck-thinning effect, but instead of losing life, you would lose tempo because the fetched lands enter play tapped.
A third reason, not mentioned by either of the other answers, is to have a way to shuffle your deck for a very low cost. This is most commonly seen in Legacy, where cards like Ponder and Brainstorm are powerful selection effects, but might leave undesirable cards on top of the deck; shuffling replaces those cards with (hopefully) better draws. Modern doesn't have either of those cards, but there are still occasional effects that allow a player to see the top of their deck such as Mishra's Bauble and Courser of Kruphix and the timing of fetchland use when those effects are available is certainly something to think about.
That said, this is probably not the case for an 8-whack deck, nor do I know of any particular mono-color modern deck that runs fetchlands specifically for that purpose. If there was one I would expect it to be mono-blue to take advantage of Jace, the Mind Sculptor's effective Brainstorm, though I haven't done an exhaustive search of all cards which would benefit from free shuffling.
In addition to the excellent answer already posted, there are times that a deck wants to have multiple lands enter the battlefield. This particular deck does not benefit from this, but there are others that do.
Any deck with the 'Landfall' mechanic will want to have the ability to drop lands, and instant speed is a big plus, for example to activate Lotus Cobra's ability. In a different Red Deck, a player might use Searing Blaze, which also benefits from Landfall.
Granted, Lotus Cobra is usually seen in multi-color decks, but the concept of Landfall is there.
A fourth reason is cards that benefit from having lands in the graveyard, such as Tarmogoyf, Barbarian Ring, Crucible of Worlds or Splendid Reclamation
The minimum deck size in Magic the Gathering is 60 cards for constructed formats, each fetch land you play effectively lowers that minimum deck size by 1 card, since you will be using that land to pull another land out of your deck right into play as if you had played it instead, at the cost of one life.
This means also that you have less land left in the deck, once you're at your right number of lands for a deck, drawing more lands means you haven't drawn anything to use the mana those lands make, slowing you down.
A third reason is in specific decks, though they are not usually mono colored, landfall. The landfall mechanic triggers whenever a land enters the battlefield under your control, using a fetch land means that effect will trigger twice, once when you play the fetch land and once when the land it fetched comes into play, allowing you to double up on some of the effects. Landfall decks used fetch lands to make more use out of cards like Lotus Cobra or Steppe Lynx in a single turn.
Another reason not yet mentioned: you can fetch utility lands. Examples are Mystic Sanctuary and Dwarven Mine. It's not common to run these lands in monocolor decks, but it's possible.
The deck you linked has none of these synergies; it is running fetchlands to thin the library.
