How do you checkmate with two rooks?
This is the only basic checkmate where your own king does no work at all. It can sit in its corner from the first move to the last.
Two rooks and nothing else can force mate against a lone king.
The method is called the ladder mate, and some people call it the staircase mate. Here it is in three steps.
1. Put one rook on the row next to the enemy king, giving check. The king has to leave that row.
2. Put the other rook on the row the king has just moved to, giving check again.
3. Keep swapping. Each check pushes the king one row closer to the edge.
The last check comes when the king is on the edge row and has no row left to run to. That is checkmate.
Why does the ladder work?
Because one rook is always doing the holding while the other is doing the checking.
A rook attacks a whole row at once. So the rook you are not moving is a wall across the board. The king cannot cross it, cannot step onto it, and cannot go round it quickly enough.
That is why two rooks need no help from your king, while one rook does. With one rook there is nothing to hold the row you just pushed the king off.
What do I do when the king attacks a rook?
This is the only thing that can go wrong, and it has a one-line answer.
Slide the attacked rook along the same row, as far from the king as you can. Usually that means all the way to the h-file or the a-file.
The wall stays exactly where it was, because a rook holds the whole row wherever it sits on it. The king has gained nothing and you have lost nothing.
The move that loses is giving the check anyway. If the enemy king captures a rook, you have thrown away half your force. One rook and a king can still mate, but the job takes far longer.
Which way should I push the king?
Towards the nearest edge. Look at the enemy king and ask which edge is closest: the top, the bottom, the left or the right.
If the king is on the fourth rank, pushing down to the first rank takes three checks. Pushing up to the eighth takes four. Pick the short way.
You can also push sideways, towards the a-file or the h-file, by putting the rooks on files instead of ranks. The ladder is the same idea turned ninety degrees.
Can I stalemate with two rooks?
Yes, and it is the one way to lose half a point here. It happens when the king is on the edge and you then make a quiet move. That move can take away the king's last square.
The ladder mostly protects you, because every ladder move is a check, and a player in check is never stalemated.
So the rule is simple: while you are laddering, check on every move. The only quiet move you ever play is the safety slide. By then the king always has a square to move to.
A drill for the ladder
Put two rooks and your king on one side, a lone king in the middle for the other, and mate it in under ten moves. Then do it again pushing the king to a different edge.
Then make it harder. Move the lone king next to one of your rooks on purpose, and practise the safety slide until it is automatic.
Most learners can mate from anywhere in eight or nine moves after twenty minutes of this.