How does 20Q work? How do the program writers start to build such a thing?

Its based on Artificial Neural Networks, which are programmatical methods of trying to duplicate the activity of the human brain. It comprises of a learning stage, where the paths to different methods are chalked out. Then, when the input comes in, the program tries to track the path of the successive answers to ones that match the user's input closest (one method of achieving this is finding the least difference between each pair of input and entry in the database). This is done stage-wise, until a good guess is made on the possible solution to the user's responses.