Black’s king is in the middle of an empty board, White’s king is asleep on e1, and it is mate in ten. Play the line through on a real set before you read the explanation.
a 
e Ten moves, and Black was free to run anywhere he liked. Sixteen is the worst case from any legal position, so the fifty-move rule is never in play. What kills beginners here is not the method. It is impatience: they check, the king runs, they check again, and nothing has changed after twenty moves.
Cut the board in half, then keep cutting
1.Ra5+ is not really a check. It is a fence. From a5 the rook owns the whole fifth rank, so the black king is now locked into the top three ranks and can never come back down. That fenced area is the box, and the entire ending is the box getting smaller, one rank at a time.
The rook alone cannot shrink it. Look at the line again: moves two, three, four, six, seven and nine are all king moves. The rook moves four times in the whole game, counting the mate, and every one of those moves shrinks the box by a rank.
Which edge you drive him towards hardly matters, so pick the near one. In the diagram 1.Rd1+ would do the same job as 1.Ra5+: the check comes down the d-file, Black has to step to one side of it, and the fence stands on a file instead of a rank. What decides the choice is which edge your own king can reach first, because he is the slow piece in this ending and every rank he walks costs a move.
The shoulder, and why the kings end up facing each other
By move four the kings stand on e4 and e6 with one square between them. That standoff is the opposition, and it is the trigger for the whole method: kings facing, one square apart, your move. Now 5.Ra6+ pushes him back a rank, and the fence has moved up with him.
Then Black does the only annoying thing he can do: he sidesteps. After 6.Ke5 Kf7 the kings are no longer facing. So White plays 7.Kf5, shouldering across, and the standoff is back with Black to move. 7…Kg7 8.Ra7+ and he is on the eighth rank with nowhere to hide.
The pattern is three steps on a loop. Face him. Check along the cutting line. Follow him up with your king. Once you have felt it three times it stops being a technique and becomes a habit.
When he will not step aside, wait
Sometimes the kings are facing and it is your move rather than his, which is the wrong way round. You do not want to check and you do not want to move your king backwards. So you play a waiting move with the rook along the cutting line: from a5 to b5, or all the way to h5, whatever keeps the fence intact. Black has to move, he gives ground, and you carry on.
This is also the answer when his king walks up and attacks your rook. Slide the rook to the far end of the same line. He has gained a tempo and lost his position, which is a bad trade for him.
One more habit that saves a lot of time. Decide at the start which edge you are driving him to, and then do not change your mind. Beginners cut on the fifth rank, get excited about the a-file, swap back to ranks two moves later, and the box never actually shrinks. Pick the fence, hold it, walk your king.
a The stalemate that steals the point
Here is the position I have watched more than one junior reach with a won game and a big smile.
8 White’s king went to f7 one move too early. The black king on h8 is not in check, g8 and g7 belong to the white king, and h7 is covered by the rook on the seventh rank. No legal move, no check, draw.
The fix is a habit, not a rule to memorise: before any quiet move, count the lone king’s squares. If the answer is zero and you are not giving check, play something else. In that exact picture White should have played Kg6 rather than Kf7, when the black king must go to g8, and Ra8 is mate next move. There is more on the same trap in king and queen against king, where the queen makes it far easier to blunder into.
Set a rook and two kings up right now and mate from the centre five times in a row, once with a friend defending as stubbornly as they can. Then do it with two rooks, which takes about four moves and no thinking at all, and go and solve a handful of rook endgame puzzles to see the same king and rook coordination in positions with pawns still on.