Sensing light levels is quite simple. You connect the light dependent resistor in series with a known constant resistance, connect one end to ground and the other end to your high logic level (5V), and the voltage at the junction varies with brightness. Connect the junction to an analog input to read the voltage, and now you have a number in your sketch that varies with brightness. It's trivial to test whether that value is above or below some constant threshold value and change the state of an output pin based on that. At this point you have an output pin which changes state according to the brightness - I suspect that's essentially what the Maplins circuit does.
Because the I/O pins have internal pull-up resistors which can be enabled by software, you can use the internal pull-up as your known constant resistance. This simplifies your circuit to an LDR with one end connected to an analog input, and the other end connected to ground.
I don't know what would be involved in activating the clock, but bear in mind that the digital I/O pins only handle 20 mA safely so if you need to control more current than that you'd need to have some sort of driver circuit between the Arduino and the clock.