For example to break Shadow Knight loops (which was a really powerful combo in CotC), counter Soul Reaper, Eternal Renewal… and more recently Avatar of Twilight, or location counter with Void Wretch.

It is also used for balancing purposes with similar intention as before to break infinite recycling: Hunter's Gambit and Mimic for example. (source)

Since Lost Lands the exile feature has gained a whole other meaning. Currently exile can be used:

• To buff allies: Rotling, Death Collector, Gravemonger, etc
• To regain health: Wizent's Prayer
• To "kill off" an enemy or remove an item/ability/location from the board: Aldmor Chieftan, Twilight Ritualist, Yahari: Valley of Doom, Groundshift, Loest, Savior of Layar, etc
• To damage an opposing ally: Tar-written Scroll
• To permanently remove certain cards from a match after use: Mimic, Hunter's Gambit
• To damage an opposing hero: Void Wretch
• To send an exile card to your hand: Racksul: Warped Desert

• Hero:

1. Loest, Savior of Layar (5SE: Target ally, item or ability is exiled)

• Allies:

1. Spirit Warden (0: Target ally in a graveyard is exiled)
2. Death Collector (2: Target ally in a graveyard is exiled)
3. Sorcerer of Endia (5SE: Target enemy item or ability is exiled…)
4. Twilight Ritualist (…target opposing ally, item or ability with cost 4 or less is exiled…)
5. Aldmor Chieftan (3SE: Target non-Aldmor ally is exiled)
6. Void Wretch (When a card is exiled, all opposing heroes take 1 damage. 0SE: The active location is exiled)

• Abilities:

1. Wizent's Prayer (Target card in a graveyard and all other cards with the same name in that graveyard are exiled)
2. Groundshift (When Groundshift is summoned, the active location is exiled…)
3. Sinkhole (Target opposing ally with cost 2 or less is exiled…)
4. Shatter Ice (All frozen allies are exiled)
5. Hunter's Gambit (…draw 2 cards and Hunter's Gambit is exiled)
6. Redux Channel (Target card in your hand is discarded: draw 2 cards and Redux Channel is exiled)
7. Migration ((…) The active location is exiled)
8. Annihilate (Target opposing ally is exiled)
9. Relinquish (Target ally has all enemy attachments exiled:…)
10. Curse of Doom (Any player 1SE: Target friendly ally is killed and this card is exiled)
11. Curse of Death (Any player 1SE: Your hero takes 5 fire damage and this card is exiled)
12. Sandblast (Target card in your hand is exiled: target opposing ally takes 4 damage)
13. Reconstitute (Target ally with X cost in a graveyard is exiled…)
14. Walk the Plank (Target friendly ally joins your crew if able or target ally in a crew is exiled or…)

• Items:

1. Splicer's Crown (Target ally in a graveyard is exiled…)
2. Tar-written Scroll (1D: Target ally in a graveyard is exiled: if the exiled ally has fire attack, target opposing ally takes 2 fire damage.)
3. Regression Chamber (1SE: Target friendly Elemental ally loses all abilities and is exiled:…)
4. Scrap Cannoneer (1SE: Target item in a graveyard is exiled: target ally takes 2 damage…)
5. Voidhammer (When your hero attacks, the active location is exiled. When a location is exiled, Voidhammer gains +1 base attack and +1 durability.)
6. Obsidian Pendant (4: Target item or ability is destroyed and your hero loses 1 shadow energy. Obsidian Pendant is exiled.)
7. Gyre and Gimble (When this weapon is destroyed, it is exiled and you draw a card)
8. Saltfang's Clawbook (1D: Target Wulven attachment or support ability in a graveyard is placed in your hand. This card is exiled)
9. The Wraith's Wake (When this ship leaves play, its crew are exiled…)

• Locations:

1. Gaderi: Braxno Citadel (2SE: This location is exiled at the end of your turn)
2. Yahari: Valley of Doom (4SE: All allies with cost 3 or less are exiled. // 2SE: Draw 2 cards and this location is exiled)
3. Irum: Lightning Quarry (itself)

• Allies:

1. Rotling (When Rotling is summoned, you may exile target ally in a graveyard…)
2. Sword Gobbler (when summoned, you may exile target weapon in a graveyard…)
3. Gravemonger (…you may exile target card in a graveyard and…)
4. Rust Weevil (When Rust Weevil is killed by combat damage from a hero, that hero's weapon is exiled)
5. Vozitian Saboteur (when summoned, you may exile target item or ability with cost 2 or less…)
6. Darkwood Leech (When killed or discarded from anywhere, your hero heals 2 damage and Darkwood Leech is exiled)
7. War Giant (When War Giant deals combat damage to an ally, that ally is exiled)
8. Crusher of the Weak (when summoned, target opposing ally with cost 3 or less is exiled)
9. Winged Redeemer (when summoned, target enemy support ability is exiled…)
10. Coran, Darkhand (When an opposing ally is killed in combat, its controller exiles the top card of their deck)
11. Nitish, Darklight Mystic (1SE: Nitish is exiled and returned to play )
12. Subjugated Wisp (When Subjugated Wisp is exiled from play or a graveyard, its owner draws a card…)

• Abilities:

1. Mimic (Target ability in a graveyard except Mimic with cost 3 or less is summoned at no cost, and Mimic is exiled. If the summoned ability was not an attachment or support ability, it is also exiled)
2. Lythian Sledgehammer (killed ally with this weapon)
3. Soul Reaper (The bottom 5 allies in your graveyard are exiled. Your hero heals 2 damage for each ally exiled this way. Soul Reaper is exiled.)
4. Eternal Renewal (Shuffle the cards in your graveyard into your deck. Eternal Renewal is exiled)
5. Hit and Run (At the end of your turn, Hit and Run is exiled and the ally it was attached to is shuffled into its owner's deck)
6. Serenity Now (… 1: Your hero heals damage equal to the number of allies you control and Serenity Now is exiled)
7. Kraken Attack (… If a ship was targeted, the lowest cost ally in its crew is exiled;…)

• Items:

1. Snare Trap (The next opposing ally with cost 3 or less that is summoned is exiled)
2. Lythian Sledgehammer (When your hero kills an opposing ally in combat, that ally is exiled…)
3. Gemstone Javelin (Gemstone Javelin is destroyed, or exiled if its base attack is 2 or greater)

• Locations:

1. Arthyle's Pass: Dead Man's Breach (When an opposing ally with cost 2 or less is killed, that ally is exiled…)
2. Racksul: Warped Desert (0SE: The next card owned by an opposing player to be exiled this turn is placed in your hand)
3. Shold: Tempest Gorge (When an item or attachment is destroyed, it is exiled)
4. The Wilds: Hall of Doors (1SE: Target friendly ally with cost X is shuffled into its owner's deck: seek an ally with cost 1 + X: that ally is exiled and it enters play under your control at the end of your turn)