When the 555 fires, it brings VPAD1 low, and a counter starts counting horizontal lines. When it reaches 15 (starting from 0), it brings VPAD1 high again. So, a paddle is 15 vertical lines high.
A second potentiometer, "Paddle 1 Control" is used to calibrate the circuit so that the paddle's position ranges from the top to the bottom of the screen.
The original game had a bug where the paddle wouldn't go to the top of the screen. This seems to be related to some non-ideal behavior in the 555 timer chip. This simulation has a 555 timer that stops immediately when the threshold voltage reaches the control voltage. But when simulating with a more accurate SPICE model of the 555, that doesn't happen if the control voltage is too low (less than about 1.5V). The timer won't stop quickly enough in that case, which may explain why it was impossible for the paddles to move to the very top of the screen. (A longer delay means the beam will be too far down on the screen when the paddles are drawn.)
Next: Drawing the Paddles.