The flying general
A rule that reads like a restriction and plays like an extra piece
Updated
The two generals may never stand on the same file with nothing between them. Any move that would leave them facing each other is illegal — including moving away the last piece that stood between them. The rule itself is on the rules page, and the endgame page calls it a weapon. This page shows the two things it does on the board, on positions checked against the engine.
A general looks all the way up its file
The simplest way to read the rule is as an attack. A general looks straight up its file, as far as the first piece in the way, and the enemy general may not stand on anything it can see. Everything below follows from treating the general that way: a piece that can only move one step inside the palace also covers a line that runs the whole length of the board.
The attack is mutual. If your general can see the enemy general, the enemy general can see yours, which is why the rule is written as a ban on facing rather than as a capture. You can never create that position; the question in every game is only what it stops the other side from doing.
The piece in between cannot move off the file
When the two generals share a file and exactly one piece stands between them, that piece is all that keeps them apart. It may slide along the file, but any move that takes it off the file would open the line, so every such move is illegal.
So a piece standing between the generals is not only unable to leave; it is a target that cannot run. Chariots, cannons and soldiers can at least move along the file, but a horse, an advisor or an elephant can never keep to a file when it moves, so any of them standing there alone is frozen outright. When you see the enemy's last blocker, ask what attacks it.
A general guards a whole file
Because the enemy general may never step onto a file your general can see, your general takes away every point on that file inside the enemy palace, from the far end of the board. In a mating attack that is often exactly the square the attack was missing.
Nothing on the board attacks f0. The only thing covering it is a rule, applied by a piece nine ranks away. This is why an endgame general that has walked to an open file is doing the work of an extra piece, and why a mating net that looks one attacker short often is not.
What to do with it in a game
Put your general on the file the enemy general needs. Once the board thins out, a general one step to the side can take a whole file away from its opposite number. It costs one move and no material.
Before moving a piece off the generals' file, count what is left on it. If it is the last piece between them, the move is not available, however good it looks. The engine here will not let you make it, and neither will an arbiter.
Treat the enemy's last blocker as pinned, because it is. A horse, advisor or elephant in that spot cannot move at all, and a chariot, cannon or soldier can only move along the file. Attack it, and it cannot get out of the way.