Take a look at the games played by well known players as Capablanca, Fischer and Carlsen. Notice that many of those games stretch beyond 40 moves in length. Even at this high level, endgames play a key role in who earns the full point. The same trend becomes even more evident at lower levels. While there seems to be a general obsession about opening theory, it is the endgames that carry the highest weight in terms of scoring victories and gaining rating points.
Winning a pawn in a combination in the middle game can be trivial. Converting this pawn into a full point can be a long and hard journey. The purpose of the endgame is to convert the material advantage into checkmate. This is why every pawn can promote into a queen. The point is to use this queen to make a checkmate. And that is when the game comes to its victorious conclusion.
The endgame begins when enough material has been exchanged. It starts at around 1 rook + 1 light piece + 0 or more pawns versus 1 rook + 1 light piece + 0 or more pawns. Or it starts at 1 queen + 0 or more pawns versus 1 queen + 0 or more pawns. The kings should be able to walk out into the center of the board without risking to become checkmated.
It would seem natural to begin mastering the endgame by learning the King+pawns versus King+pawns endgames. These endgames are characterised by having to calculate long variations of different pawn races. Also about knowing typical pawn structures and rules that help to decide who has the advantage in a certain position. Then comes the King + 1 knight + pawns versus King + 1 knight + pawns. This endgame is considered to be similar to King vs King endgames, with the knights adding extra flavour of pawn grabbing and helping to promote passed pawns and stopping the opponent's passed pawns.
The endgame that can earn you many new victories seems to be the King + 1 rook + pawns endgame, also known as "Rook endgames". Learning the rook endgames will both increase your chances of victory and your ability to defend these positions. This means that you can aim for a rook ending both for winning purposes and for defensive purposes.
Beginning to improve your chess by learning endgames makes sense, because it is the endgame's purpose to convert advantages into checkmates and thereby victories. Moreover, endgames are full of exciting ideas and combinations. When having a look at chess games played by great masters, one will see the beauty of endgames and hopefully get inspired to explore them.