I personally do not use TDD, but one of the biggest pro's I can see with the methology is that customer satisfaction ensurance. Basically, the idea is that the steps of your development process are these:
1) Talk to customer about what the application is supposed to do, and how it is supposed to react to different situations.
2) Translate the outcome of 1) into Unit Tests, which each test one feature or scenario.
3) Write simple, "sloppy" code that (barely) passes the tests. When this is done, you have met your customer's expectations.
4) Refactor the code you wrote in 3) until you think you've done it in the most effective way possible.
When this is done you have hopefully produced high-quality code, that meets your customer's needs. If the customer now wants a new feature, you start the cycle over - discuss the feature, write a test that makes sure it works, write code that passes the test, refactor.
And as others have said, each time you run your tests you ensure that the old code still works, and that you can add new functionality without breaking old one.