By building part of the software and demonstrating that it works, we're able to complete one of the most difficult types of testing: validation.
I can't remember when I first heard the words verification and validation. It seemed like such nonsense to me; the two words sounded like synonyms. Now I know the distinction, which is important. Verification means it conforms to specification; in other words, the software does what you said it would do without failing. Validation means that the software is fit for use, that it accomplishes its intended purpose. Ultimately, the software has no value unless it accomplishes its intended purpose, which is a difficult thing to assure until you actually use it for its intended purpose. The best person qualified to validate the software is a person who would eventually use it. Even if the target users can't tell me conclusively that it will meet its intended purpose, they often can tell me if it won't, as well as what I might change to be more certain that it will meet its intended purpose.
Building software in small pieces and vetting those pieces early allows us to begin to validate sooner. Inevitably, this sort of validation results in changes to the software. Change comes a bit at a time, or it arrives continuously throughout an agile development process, which agile people believe is better than getting it all at once in a big glut at the end of the project, when time is tight and tensions are high.
Being a tester in an agile environment is about improving the quality of the product before it's complete. It also means becoming an integrated and important part of the development team. They help ensure the software—each little bit that's complete—is verified before its users validate it. Testers are involved early to help describe acceptance criteria before the product is written. Their experience is valuable to finding issues that likely will cause validation issues with customers.
At the end of the day, an agile tester likely will pour over the same functionality many times to verify it as additional parts are added or changed. A typical agile product should see more test time than its non-agile counterpart. The uncomfortable truth for testers in agile development is that all of this involves hard work and a fair amount of retesting the same functionality over and over again. In my eyes, this makes the tester's role more relied on and critical than ever.