To "show" something means that you display it so that everyone can see. Here is a short story about the invention of flying machines... "I have an idea on how to make a machine that will fly in the air." "That is impossible. How can a heavy machine fly?" "I think we need to shape the wings so that they will push the machine up into the sky, we will also need a powerful engine to push the machine forward into the wind." "Your idea is crazy and it will never work." But, the inventor didn't bother arguing with the skeptics. The machine was built and a test was arranged. It worked!! The machine flew in the air!! There was no need to argue with the skeptics. All you need to do is SHOW them that it will work. "I'm going to show that it works!" - this is talking about the future. "Shown to work" - this is talking about the past.

It means that his theories proved to be correct. His theories worked.

If something is "shown to work," its functionality or efficacy has been proven by experiment or observation. For example, if an economist suggests a theory (of pricing options, for example) and that theory correctly predicts market behavior (options prices more or less match what the model predicts), that theory has been shown to work. It does imply that someone has tried using the theory and found that it was correct, but the emphasis is on testing the theory and finding that it is correct. If I wanted to say only that I acted on a theory, strategy, or plan, I would use "apply," "implement" or "put into practice." There is no suggestion of testing with these expressions, only of use.