How does the telephone's keypad work?

There is a conductive pad on the bottom side of the button that "shorts" out the spiral. Usually dark grey since it is carbon based.

The spiral just give you a higher probability that the pad will touch both sides at the same time.

The spirals are electrical contacts. Shorting the traces in a spiral causes the keypress to be detected.

The backs of the keys have a layer of conductive rubber or plastic. Pressing that layer against the spirals closes the circuit just like you had use a piece of wire to make the connection.

The black rubber on the back of the keys is actually somewhat conductive and provdes a current path between these swirls once a key is pressed.