Many decks in Magic are built around a few cards that will eventually win them the game. Whether they are long term value cards, combo bombs ready to go off, or just otherwise hard to deal with cards, decks often need this lynch pins in order to do well. If someone were to answer these threats, well, the term backbreaker comes to mind.
Surgical Extraction is that backbreaker.
Look at this pinnacle of efficiency, sporting some of the coolest, simplest, and most haunting pieces of art in the game.
Why you want this in your sideboard
The reason this card sees so much play (it's current number 37 for cards that see modern play) is that it is a great sideboard card. It helps hard counter certain deck archetypes. It can hit so many things.
Against Dredge, or Hollow One, where your opponent's threats live in the graveyard, it can take out their Bloodghast's for just 2 life, creatures that would otherwise chip you for tons of damage over the course of a game.
1 Phyrexian mana is basically a free cost, since two life is a small cost to pay for such an effect.
Against something like Through The Breach decks, it can act as a real combo breaker. Taking out a Griselbrand in a graveyard that is just waiting to ruin your day, and cutting off your opponent from doing it again, is a great feeling. Plus, at instant speed and only 2 life or 1 black, it's a great way to fizzle your opponent's Goryo's Vengeance as he tries to win the game.
In decks that are running Snapcaster, hitting two threats for 2 mana and 4 life is often enough to seal the deal on a game. It may not be what you want to rely on, but will certainly be a welcome gimping to your opponent's strategy.In Other Scenarios
Surgical Extraction also isn't too much of a dead card if you're not lining it up with the main threats you want to get rid of. Waiting for the right card to hit is often the right move, but getting rid of your opponent's fetch lands for the rest of the game isn't a bad thing at all, especially if they have a second copy in their hand ready to go. And on that subject, the ability to look at your opponent's hand AND deck on its own is sometimes worth the two life. Maybe they're playing some janky home brew (shout out to our other series), and you want to be prepared for any shenanigans they're planning to drop on you.
And as far as commitment to the card goes, it's nearly as cheap as you can get. 1 Phyrexian mana is basically a free cost, since two life is a small cost to pay for such an effect.
Other Jank uses
I'd be remisced if I didn't conclude with one of my favorite uses of Surgical Extraction, and that's in a Shadowborne Apostle combo deck. The deck works around playing a Shelldock Isle, and hiding Emrakul, the Aeon's Torn in it. Then, after having one of your apostle's get put into the graveyard, you target your own card, and Exile most of your deck. Then, you cast Emrakul and (maybe) win the game. So janky, so fun to pull off