They implement logical gates by turning on/off transcription, so the result of one gene turned on will be that another is turned off (a NOT gate). And so on.

I assume the idea is less to build an adder and an ALU ( :-) ) and more to build behaviour on environmental input.

There is a video under the "demo" tab.

Edit: Under the "Publications" you find an abstract and a good explanation. Read that instead of my copy/pasting. :-) Their basic point is "This approach leads to highly repetitive and modular genetics, in stark contrast to the encoding of natural regulatory networks"