A business plan definitely isn’t required to open your business, especially if you aren’t seeking a loan.  However, good planning can help you achieve your goals faster and with lower overall costs.
A business plan, whether you use a one-page business plan like the lean plan here or a traditional plan, forces you to formalize your idea and conduct market research that will help to further define your brand.
Often, creating a workable plan requires you to determine key metrics and ideal customer segments.  Formal plans may even ask for sales forecasts or financial forecasts to help lenders better understand your projected profits.
You won’t see as much of that with a lean plan, though the numbers-driven details should definitely play a role in your regular review once the business is up and running.