One way is through our observations. We develop beliefs and knowledge based on what we have seen. Like when the first person circumnavigated the global we all started to believe that the world was round. Before that everyone thought the earth was flat. Before circling the world, everyone's observations led people to believe incorrect facts because they didn't have a complete picture. Using this systems as the only system to establish our beliefs means that we establish our beliefs on partial information. We cannot observe everything, so we can never really base our beliefs on a complete picture. The scientific method is just an algorithm to increase our ability to observe and make appropriate conclusions. It has led to great advances in our knowledge, but it still paints an incomplete picture.
Another belief system is establishing our beliefs through feelings. If there is a God, and if He loves us, than wouldn't he try to communicate with us? I believe that he communicates with us through feelings. This can be done through our conscience, guilt, and with other feelings of hope, joy, and peace. If you hear something and feel peace, than it is God communicating with us.
I just find it weird that a person who only bases his knowledge on a partial picture would ridicule others who base their beliefs on receiving light and intelligence from God. Don't get me wrong. As a scientist, I believe that the scientific method is very useful in exploring our world and developing beliefs. However, it only paints an incomplete picture. There will always be more observations to make. As an example, there is an entire research field studying the performance of computers. They still don't know the details of everything that is going on inside of a computer system. And that machine was built by humans. We don't even have a complete picture our of our personal history on this earth. How many observations have we forgotten about? How could we ever expect to know everything about this world through observations? So someone that bases his entire belief system on only their observation, and tells others that they are crazy for believing what they do, seems to be the one that is short sighted.
I know that God lives because I have felt him talk to me. I know the Book of Mormon is true because I asked God and he told me it was true. I know that many people might try to bring evidence (or observations) to try to contradict my knowledge, however they are only partial observations and only paint incomplete pictures. I think I would rather trust in what God has told me. Not only does it give me a more complete picture, but it has brought a lot of peace and happiness in my life.