If I have the right "Lightboard" in mind, these work by putting a piece of very transparent glass between the tutor and the camera. The glass has some white LEDS on the top and the bottom edges that send light through the glass (due to something called total internal reflection, the light does not exit the glass for the most part). When the tutor writes on the glass, this puts a thin layer of the fluorescent marker on the glass, and allows light to exit through where the writing is. This light reaches the camera and thus we can see the writing.
Another interesting question is why doesn't the writing appear backwards since the tutor is behind the glass. It would, if you did not do anything to prevent this! Two ways to make the writing not be backwards: (1) point the camera at a mirror which reflects the tutor/lightboard (2) digitally flip the video image. (Interestingly, if the tutor is right-handed, then it will appear as them being left-handed in the video.)