A paddle is a game controller with a round wheel and one or more fire buttons, where the wheel is typically used to control movement of the player object along one axis of the video screen (wikipedia)
Not many computers of the 80's had analog to digital converters built in, so the most common method for reading the wheel position position was to generate a pulse of a width proportional to a voltage or a resistance, and measure it using a loop counter.
The MSX computers even have BIOS and BASIC functions for dealing with paddles but such peripheral was never officially released for the system.
On the other hand Atari paddles are rather ordinary and remarkably easier to find even today (not brand new of course) wich made them a perfect choice for use in MSX as long as the differences between the systems are properly addressed: The Atari system requires only a potentiometer, while the MSX requires a timer chip, and that defines the design of the adapter circuit.
The circuit is based upon a CMOS precision monostable C4538. Rather than a pair of 555's this chip can be triggered by a positive pulse, as generated by MSX, and allow the use of high resistance components in its RC network, like the 1MOhm from Atari Paddle potentiometers.
The capacitor is calculated to match the MSX timing:
Maximum counts: 255
count each: 12us
Tmin = 12us
Tmax = 12us*255 = 3,06ms
According to CD4538 datasheet T=RC Hence 3,07ms = 1MOhm x C -> C = 3,25nf being 3n3 the closest commercial value.
The board was specially designed for the return of the Square Inch contest and measures - of course - 1x1 inch