Agile approaches are very much about adapting and using short increments when dealing with uncertainty. Starting a new company comes with a lot of uncertainty. Agile principles can help reducing risk and improving the odds of success.
Eric Ries is the inventor/initiator of the “Lean Startup” movement. His concept is to incrementally develop your customers and your business. By running experiments, some people also call them spikes, you learn a lot about the business idea you are working on. By using small increments you avoid spending months or years building a product or offering a service that in the end is fantastic but nobody is willing to spend money on. Eric Ries helps you de-risk your business idea.
A complementary technique is customer development a term coined by Steve Blank. While product development important (your product can also be a service) it is equally important to also develop your customer base. In his book “Four Steps to the Epiphany” he describes this process in details. His book is a work book, so be prepared that you will have to do homework.
Why am I mentioning these two? The work of both of them is based on agile principles. Therefore if you are considering to test one of your business idea then the work of these two authors should be part of your preparation. And both are not limited to new companies. If you are tasked to build that brand-new product in your company then you are basically running a start-up. The only difference is that you run it in the context of an existing company, which can be beneficial (e.g. financial backup) or a hindrance (e.g. bureaucracy).