Pressed reads HIGH

Up is LOW

With the key up, the switch is open and BUTTON is connected to one thing on the board: R7, 10 kΩ, to GND. No current flows through it, so there is no voltage across it, and BUTTON sits at GND: 0 V, LOW.

Without that resistor BUTTON would be joined to nothing while the key is up. A pin like that floats: it picks up whatever is nearby and reads HIGH and LOW at random. The resistor is what makes "up" a definite answer. It is called a pull-down because it pulls the line down to GND when nothing else is driving it.

Down is HIGH

Press the key and the switch joins BUTTON to VCC. The resistor is still there, but a switch is a far stronger connection than 10 kΩ, so BUTTON sits at VCC: HIGH. The only thing that changes is a small current through R7, VCC divided by 10 kΩ, for as long as you hold the key.

So in a sketch:

pinMode(BUTTON_PIN, INPUT);        // the block has its own pull-down
if (digitalRead(BUTTON_PIN) == HIGH) {
  // the key is down
}

The old page had it the other way

The page this handbook replaces said the key "outputs LOW when pressed", and its sketch set INPUT_PULLUP and treated LOW as pressed. On this board that is backwards: it reports a press whenever the key is up. Pick "LOW (the old page)" in the figure to watch it happen. Many buttons are wired the pull-up way, which is why the mistake is so common; this one, like the TK04, is not.