Anyone who has studied some issue in even moderate depth can immediately relate to the experience described in this short verse. You think you know it all, until you sit down and read some books from authors who don't agree with you. Then you suddenly realize that the people who disagree with you are not hopelessly irrational or simply blind to reality, but really have thought out their opinions in some detail. More than that, they have noticed sides of the story about which you had never been aware, in light of which the contrary opinion suddenly gains a new light of plausibility.The one who first states a case seems right, until the other comes and cross-examines.
Only at that point can a person truly say that she has thought deeply about some issue or other. Until she has come face to face with a contrary perspective and considered it fairly, on its own terms and in its own light, and attempted in this way to recreate within herself the experience of those who believe this contrary opinion -- until she has done all this, it seems to me, her treatment of whatever issue is still immature and lacking.
Now, as the proverb insists, considering an issue from two sides can often leave one in a state of uncertainty and ambiguity. That's what is evident in the proverb: everything sounded great until the first person's case was examined by the second, and now it's not so obvious what the truth is anymore. What should a person do who finds herself in that ambiguous state? What conclusion should she draw from the matter?
Some philosophers say that it would be better in that case to abstain from judgment; others suggest that it would be permitted to take a side, but with a dose of humility and recognition that others who disagree are generally not wildly irrational or simply blind to the facts. In my opinion, in some cases we cannot abstain from judgment one way or another: abstinence is the same as denial. For example, a person cannot in principle remain neutral on the question of God's existence: not to believe one way or the other most often translates into acting as if God did not exist, making it tantamount to atheism. But in other cases, it would seem right not to make a judgment (or at least not a very firm one) in response to a situation of such ambiguity and uncertainty regarding the truth of the matter.
Why is this such an important verse? Because too many people don't realize that things are never as certain and as clear as they seem. I will give an example from my own personal experience. In the Romanian Pentecostal community in which I was raised, the iconodulism of the Orthodox is obviously idol worship and a violation of the commandment against the worship of graven images. But the iconodule arguments pro are quite sophisticated, and show that the relevant scriptural issues are not as clear as they might initially seem. But so few -- so few, in fact, that I don't know of any -- of those who confidently denounce icon veneration as idol worship have read any favorable treatments of the issue, such as the treatises of John of Damascus.
This happens in theological circles of other sorts, too. How many young, restless, Reformed types do you know whose theological consumption is limited to the familiar group of authors and speakers (e.g., John Piper, R.C. Sproul, etc.)? They would be hard-pressed to name very many Church Fathers or non-Protestant theologians, and in fact, their constant exposure to more polemical, less ecumenical Reformed theology might even oppose them a priori to any such reading. And the same might certainly be true in my own case, as well: I mostly read Patristic theology, or else Catholic and Orthodox writers who draw from Patristic theology; I don't exactly spend much time digging into Calvin or Luther in greater detail. And yet I have the impression about myself that I have an informed opinion about Protestant theology! Things need not be the way I consider them to be.
Uncertainty and ambiguity is a defining mark of human existence. Lots of things are simply not as clear as they could be, and a person who has read widely enough on some particular issue will admit as much. There is important spiritual wisdom to be gained in recognizing this and refusing to live in the black-and-white Talibanish mentality of some for whom no questions or doubts are even possible.