One way to answer the question (and I suspect there are many approaches) would be to define reason as the product of an instrument of rationality, i.e., a mind. What makes the question more than trivial for us, however, is that our minds are connected to bodies that interfere with reason. Emotion, instinct, and biases often dominate the function of the mind and produce various type of irrational thought.
An oddity of the human mind is that when it considers its own thoughts, it usually finds a great deal to like and when it considers the thoughts of others, it often is skeptical. I've found this can happen even when I'm in agreement with the thoughts of others! (I believe the Ontological argument is valid and so does my friend. But I think he might be wrong.) We are often over-skeptical of the ideas of others and over-confident of our own ideas.
Since every mind will be disrupted by a unique set of interference, one solution to the problem of knowing if an explanation is rational or not is to submit it to a collection of minds in some public forum (such as this one!) where it can be analyzed. Presumably the consensus of many minds will eliminate the most common forms of bias. Problems with this approach are left as an exercise for the reader!
Often we work together to form a set of rules we agree on and can apply to validate our reasoning. Remarkably, there are many overlaps between one set of rules and all other sets of rules. Therefore, over time, we've learned to use these basic rules of reason and declare that thoughts which conform to these rules are rational. You can find examples of these rules in the other answers.