You are lumping together sensory signals (feelings, sights, sounds) with the internal decision making of your brain (beliefs).

I think you are confused by the literal definition of belief, because as it is defined, a belief is anything you think is true, specially without literal proof. There's tons of stuff we believe are true without certain proof: math, science, memory to a certain extent...

Let me explain, not everyone is a mathematician, not everyone is a scientist, especially an absolute expert in every field. What people usually do is they it trust in people that are experts and they believe that what those say is true because they have proven it to another collection of authorities, and we believe that what those authorities are trustworthy because X. There's papers about it and I can looks them, but I and most people probably won't because I decided to believe in their words. That core belief on science, on the fact that the science as we know it is correct (which is again dependant on the interpretation of the universe we currently hold being true which we don't know...), is something people don't offer recognise as a belief, but it is.

What you believe to be true will affect how you process what you see, what you hear, and how you ultimately act.

It's important because it's the deciding factor of almost every actions you consciously take.

Let me say that I'm not, in no way shape or form, saying belief in a religious tone. I'm using the literal definition of belief of the dictionary.

For reference because I won't probably return to this comment, they way I see beliefs is similar to the philosopher Ortega y Gasset.