I drifted away from Hearthstone about a year and a half back. It was more interesting to me when it had fewer cards because it was easier for me to come back and dip in for a few rounds. Now it's like Netrunner - I don't know what anything does or what might appear at any given moment so I sit there with my basic mage deck and sometimes turn something into a sheep (the Netrunner equivalent is staring at my runner deck and trying to work out why I thought Crypsis was going to come in useful here). Now my main source of interest in Hearthstone is looking at little deck experiments where people try to break the game or create weird scenarios by exploiting card interactions. This is just such a thing - look!
The massacre comes about because of two cards; Knife Juggler and Dreadsteed. Knife Juggler has been about for ages. The important thing about it in this instance is that it deals damage to a random enemy after you summon a minion. Dreadsteed is a new card from the Grand Tournament set. It only has 1 hit point and when you kill it (provided it hasn't been silenced or whatever) it summons another Dreadsteed.
The setup showcased by The Optimistic Brit (I see what you did there) has each player setting up their side of the board with five Dreadsteed and two Knife Jugglers. The Knife Jugglers have incredibly high hit point values because the players used other cards to boost the health of one, then summoned duplicates using a Faceless Manipulator.
When the final Faceless Manipulator is played and summons that last Knife Juggler it triggers the long pony-knifing turn. It's a pretty simple interaction, actually. The Knife Juggler appears and it triggers the other Knife Juggler on that side of the board to fire off its random damage ability. The random damage hits a Dreadsteed on the other side. The Dreadsteed is killed but, as per its special ability, it just gets replaced with another Dreadsteed. The replacement pony appearing triggers the Knife Jugglers on that team to fire their random damage and some of the original player's Dreadsteeds get killed. As long as Dreadsteeds keep being killed and replaced the action continues.
The reason it eventually ends is not that both players are dead (they're both well and truly destroyed a fair while before the turn ends). It's that the random damage is gradually chipping away at the health of the Knife Jugglers as well as hitting the Dreadsteeds. Eventually there are only two Knife Jugglers on the board, one on each side, and at 16:13 the random damage from one hits the other. No Dreadsteeds are killed and so the cycle of summoning and damaging finally ends.