It's always useful to provide a
link so that people don't have to do extra work to determine what device you're talking about.
A link to the
manual would also have been a nice touch.
Also some description of the joystick (as per the above) would be essential.
The interface device you have can only handle digital inputs, so I hope your joystick is digital and not analogue -- not that it's impossible to interface to an analogue joystick, but it's a lot more hard work with the USB interface you have.
If it's digital and has a common connection for all switches, connect this to a voltage source (say 5V) and connect each of the switch outputs to a different digital input of the interface. Obviously the power supply ground needs to be connected to the ground connection of the card (you really only need to connect a single ground connection). Then just follow the instructions to read the inputs and do *whatever* with the results obtained.