are you sure about that? the king could have moved back up, but in front.. moved from right side to left side between those two lines, without moving in front of the queen line.
If Anand played the king to the eighth rank instead of to h6, Carlsen would have kept checking on the seventh and eighth ranks, with a draw by perpetual check.
It was the best option in both cases. He was in check when he took the queen, so the queens had to swap or black would lose his queen. At the end Anand traded his knight for the pawn as a king and knight (black's material) can't win, but a king and pawn (white's material) can, so it is in blacks best interest to swap.
what rule is that? are you implying even if only white king, black king, and black knight would be left on board, it would be a draw/win for white by default?
Instead of trying to isolate white king into a corner for checkmate?
Combinations with insufficient material to checkmate are:
king versus king;
king and bishop versus king;
king and knight versus king;
king and bishop versus king and bishop with the bishops on the same colour.
- wikipedia
Then you have the situations where a check mate is possible but cannot be forced. King and two knights against king cannot force a check mate, but if the player with the lone king knows what they're doing and wants to get check mated, they can help the other player to win. If I'm not mistaken, that counts as a draw even if the clock flags on the player with less material.
You can't checkmate with just a king and knight. Isolating into a corner won't help, as the king can not get close enough to keep the opposing king "in place", and while the knight can try to keep the king cornered, there would be no positions where the knight can check where the checked king can't escape the check by safely moving towards the knight.
EDIT: There would be positions where you don't try position the king to try to hold the opposing king in a corner where the opposing king can't move towards the knight to escape a check, but in those positions it will be able to move away in other directions.
It's not in the written rules of the game, but is a consequence of the rules.
To place the opposing king in checkmate, you must have sufficient pieces to threaten the square the king is on, and all adjacent squares, simultaneously. This means anywhere from 4-8 squares depending on the position.
King and knight, alone, cannot threaten enough contiguous squares simultaneously to cause checkmate in any position, largely because the knight must place the opposing king in check (a king cannot get close enough), but due to the mechanics of the knight's move there will always be at least three adjacent squares not threatened by the knight at that moment, and at least one that cannot be threatened by the knight's king.
You should feel free to try this yourself: place a king of one color on the board, and a king and knight of the other color, and then use only those pieces to try to construct a checkmate. It simply cannot be done. The best you can do is isolate the lone king in a corner and cause a draw by depriving it of moves (but it will not be in check at that time, so it will be stalemate, not checkmate).
And as luiz-pv9 points out, king+knight is not the only "insufficient material" combination.
He took queen with queen, clearly, knowing it would be lost to king, and then again the pawn with knight. He had a knight, of all things!