Looks interesting.
Joysticks usually have potentiometers connected to the stick to signal movement, so I'm not sure how this would work with just switches.
How does the schematic marry-up to the physical implementation?

Most joysticks actually use variable resistors on the x/y axis to check joystick position. This probably uses something similar to the Wii to detect x/y position. Most likely the KT3615 labeled in the device with switches on either side as a kill zone to stop the joystick from giving x/y coordinates off the scale. The other switches look like they go in the fingers of the device, and you bend them for different buttons. It kind of resembles the power glove, but instead of using IR for position tracking, (similar to the light gun) some kind of gyro or other tilt sensor.

The transistors in the schema are probably used to provide a constant current source (the "standard" PC gameport actually reads current trough a variable resistor, rather than a voltage). The two buttons on either side work as short to 0k impedence or 100k impedence to mark the two extremes of the X or Y axis. Most older gamepads use the same setup.