Am i missing something obvious here ?
If you need to supply a voltage that varies between zero and ten volts to the vellman board you need some way of getting an analogue value out of the arduino. Digital pins will provide you with only zero or five volts. However the PWM pins will provide you with a variable ratio mark / space (high / low ) signal. If this is smoothed with a resistor and capacitor this signal will be converted into a variable DC voltage between zero and five volts.
Now this is half the range you want. You could amplify it with an op amp but you get into all sorts of problems driving an op amp close to the supply rails. So the simplest thing is to put it through a transistor (with a pull up on the collector to 10V) so that you get a variable ratio mark / space signal between zero and ten volts. Then smooth that and you get your variable DC voltage between zero and 10 volts.
If you only want on / off control then any digital output pin with a transistor will do to give you a voltage of zero OR 10 volts.
Hope that is clear.