I would suggest that reason is a logical and rational way of justifying a belief. As such, it is a sort of methodology, and so is quite distinct from evidence – although evidence does play a key role when we are reasoning about the real world. It works by utilising such logical tools as deductive reasoning (e.g. all cats have 4 legs, Felix is a cat, therefore Felix has 4 legs), inductive reasoning (e.g. water has always been observed to freeze at zero degrees Celsius at sea level , therefore water always freezes at zero degrees Celsius at sea level), and abductive reasoning.
This last method attempts to find the best explanation for some phenomenon or other by considering which of the proposed explanations simultaneously best fits the available evidence, best predicts what we should expect to find if the hypothesis is true, is not falsified when we search for evidence that we would not expect to find if it is true, and is the most parsimonious (i.e. does not introduce unnecessary ad hoc assumptions). Mere consistency with the evidence is not sufficient, since I could invent an infinity of beliefs that are consistent with the evidence, but they would all be wrong.
In addition to this, the method of reason takes care to avoid a number of known fallacies and other errors of thinking or arguing that can lead to mistaken conclusions – begging the question, non sequitur, equivocation, anecdotal evidence, false dichotomy, argument from authority, ad hominen etc. When we don’t take care, we are liable to fall into one of many possible traps in our reasoning.
As to why we should hold reason in high regard, and privilege above other methods of enquiry, I would answer that it is the most consistently reliable way we have found of determining the truth. Whilst there are issues involving subjectivity, biases, cherished theories etc., these are human failings, or limitations in the evidence that we can obtain, and not a critical failing of the methodology itself. Over time, the application of reason does seem to lead to the discarding of false beliefs and a gradual homing in on the truth. There are still unresolved issues with the methodology concerning such things as the problem of induction, and what we can ever really know without reason or justification (brute facts), but nevertheless the method of reason has proved to be incredibly successful in advancing our knowledge of the world.
Some contend that faith, revelation, and spiritual introspection are more reliable ways of seeking the truth and justifying a belief, but I don’t think that this is borne out by the evidence. Beliefs generated and held by revelation and faith cover a multitude of conflicting theories about the creation, evolution, structure, and future of the universe and ourselves, and of morality. However, as they contradict each other in fundamental ways, they must necessarily all be wrong with the possible exception of one of them. But, since they are held to be absolutely true by means of faith, and are off limits to reason, how can we ever investigate and determine which, if any, of these ideas is true? This method of truth seeking therefore seems to lead to a huge variety of conflicting ideas, with no agreed way of confirming or rejecting them, and believers of each rejecting all the others - an impasse. Whilst we might be missing out on some great truth here, the fact that the religious landscape is one of confusion and contradiction means that we have no reliable way of knowing which this true revelation is, if any.
By contrast, when we hold a belief based upon reason, we can explain why we think the premises to be true based upon our evidence, and demonstrate that the argument is a sound one. The appearance of new evidence would perhaps cause us to revise or reject our argument, and hence our belief. Others can examine our evidence, and our reasoning, and either agree with the conclusion, or explain to us why we have gone astray. Hence, this way of seeking the truth is open to revision, and is therefore error correcting.
Some may argue that intuition is a reliable way to seek the truth. However, this is contentious. It does seem that, under certain circumstances, the processing that the brain does behind the scenes when we intuit leads to rapid and correct judgements. Also, our intuition may sometimes give us with a good direction in which to take our more formal reasoning. However, our intuition can also be highly unreliable. For example, studies have shown that the face to face interview is a poor predictor of how well an employee or student will perform. Computer programmes can do far better than this when they choose based upon a set of fine-tuned preferred criteria and a large range of relevant candidate data. Psychologists such as Richard Wiseman and Stuart Sutherland have studied and written about this at length.
Of course, this does leave open the question of why we should seek the truth at all. I think that there is probably no knockdown argument for this, but there a few possible answers. Perhaps having true beliefs helps us to improve the lot of humankind? After all, the advances that have happened in medical science have helped to prevent and relieve a huge amount of suffering. Before this, we really had no reliable way to deal with illness. Furthermore, if one has unjustifiable beliefs (either metaphysically or morally) then perhaps one is more open to committing or supporting the committing of atrocities. I think that Voltaire said: "Those who can make you believe absurdities can make you commit atrocities".
So, in general, I am inclined to think that cherishing and seeking truth (or, at least, belief justified by reason) about the universe, ourselves, and the human condition tends to lead to an improvement in conditions for humanity. Beyond that, there is the rather more tenuous idea that we find ourselves at large in a mysterious universe, so perhaps we have some sort of an obligation to try to understand it. Having said all of this, I think that one can make a good case that there is some knowledge that might be so dangerous that we would be better of not knowing it.
For example, I think that one could make a persuasive case that the knowledge of how to make nuclear, biological, and chemical weapons is so dangerous that we might have been better to have never acquired it.
One could object to this assertion in the following way:
1) This knowledge has also given us very useful and benign spin-offs that have been beneficial to humanity.
2) It was inevitable that we would acquire this knowledge anyway.
However, if this knowledge causes humanity to wipe itself out (as well as most or all of the other animals on the planet), as it may yet do, then I think that this would qualify as knowledge that is so dangerous that we would have been better off not knowing it.
I think it is clearly the case that humanity's inquisitiveness, ingenuity, and intelligence have enabled our knowledge and technology to increase at an exponential rate. However, any corresponding reduction in our belligerence, territoriality, xenophobia, irrationality, and our desire to control and dominate others has been far less marked, or perhaps negligible. This might yet be a deadly combination for humanity. To paraphrase Sam Harris, I think that humanity is approaching a bottleneck, and it is not at all clear that we will get through it.