While representing large private equity sponsors, I have seen buyers win deals against all odds and lose deals when they should have won them.
The ones who won deals almost always managed to make the seller feel warm and fuzzy. Doing this starts with understanding the psychological state of the seller. Most sellers are deeply connected to their business. They spend more time with the business than their family, hobbies, and friends, combined. When they look at their business they feel the years of sweat and blood.
Thus, and this is important, many sellers are ambivalent about selling. This is true even if they list their business for sale. Often this signifies a life-stage transition seller is not ready for and a change into an uncertain next phase of their life.

When a buyer sees the target business they see it in its current state, but in order to make a seller feel comfortable, the buyer needs to see it from the seller's perspective, recognize their mental state, and hand-hold the seller.
Each buyer has its own ways, but here are three I have seen work:

1) Just acknowledge their position and mental state. Stating that you understand what the seller is feeling goes a long way to making them feel heard and comfortable.

2) Make sure seller knows that they are impacting your life and the life of your family by selling to you. Find ways to talk about your journey and dream, what has inspired you to take this path, and how you view your future life. The seller will connect with feelings of wanting to control your own destiny, dictate your own schedule, and provide for your family.

3) Make sure seller knows their legacy lives on. This can be done in concrete ways by keeping the name of the business or putting up a plaque commemorating the seller. I have a client who bought a bar-restaurant chain and when my client told the seller he planned to keep the seller’s name and put up pictures of him and his wife, I saw the seller’s eyes tear up.

We have a tendency to focus our efforts on the concrete aspects of the business, the financials, and the terms of the LOI, but if you understand the seller’s position you will be more likely to win deals.