The idea to create a minimum viable product comes from the book The Lean Startup by Eric Ries. We covered the book in my entrepreneurship course in graduate school (one of my favorite courses) and I do recommend reading The Lean Startup if you plan to launch your product to the market using a test-and-learn or agile methodology.
The concept behind developing a minimum viable product is fairly simple. Develop a pared-down version of your offering and launch it to the market before you are ready so that you can solicit feedback from your audience about how to improve your offering, while also making some money to reinvest into your business. Two key elements of the MVP methodology are to launch early with a pared-down version of your product and to launch to a smaller audience so that you can gain feedback without risking over-exposure.
In practice, the hardest part of the process is determining when you are ready to launch and what comprises your MVP. Most experienced entrepreneurs and business coaches will tell you to launch before you feel that you are ready. I agree with this advice. Launch your beta test as early as possible. By all means, create a prototype and test it before running a beta test. Make adjustments to your offering. If those adjustments are more than 40% of your product, you might even want to conduct another round of testing. If not, you might be ready for a beta launch. As a general rule... No first version is the final version. If you wait until your product is truly market ready, it's possible that you will invest too much into something that will change a lot between your first and second launch.
Launching to a small audience for a beta test is fairly easy for a small business that is just starting out. You likely haven't built up a huge following yet anyway. You can leverage the small audience that you have as your initial beta and gather feedback from those initial customers. The process only works if you are open to the feedback that you receive and willing to make timely improvements to your products and services, but if you can offer incremental improvements to these initial customers, you will probably also earn their loyalty and trust.