Do you mean the demo in the lectures where a member of the audience used a tablet to press a button that turned on and off the lamp?
For this we used a combination of a servo motor that was controlled by an Arduino with an Ethernet shield. The program to run this code was created in the Arduino IDE (which stands for integrated development environment). You could definitely do this using a Raspberry Pi. For the “Things with Stuff” projects (you can see these on the Royal Institution webpage here: http://rigb.org/christmas-lectures/sparks-will-fly/things-with-stuff) we did to accompany the lectures we programmed a Raspberry Pi to flash lights when people tweeted. You could do something like this to move the servo.
If you’re not sure about building a webpage there are people who have built platforms that you can use.
This webpage for example has a free platform and some interesting guides to building your own switches for your home which you can control from a webpage. http://homeautomationserver.com/about