The idea of extracting rules from trained networks is very popular among the believers in classical symbol processing methods where the doctrine is that people use rules not networks (and not cases). Extracting rules also gives you some ability to understand the "reasoning" the network uses to produce its answers. Besides these reasons for trying to extract rules there is work that shows that the rules you get can give better results than the entire network. Extracting rules amounts to a really extreme pruning of the network.

There is an online paper by Shavlik on extracting rules from networks.