There are two main reasons that should be primarily stated on the importance of debate. The first being the act of convincing the public against your opponent(s), and the second being, estimation of the value of ideas.
Considering the public act of convincing and persuasion, it is important to note, that the reason we hold debate is not to come up with solutions – even though some experienced debaters will elude us into this frame. Nor, is it to convince our opponent of our true and righteous worldview. Rather, it is to convince the public to our point of view and gain their support. This is fundamentally the reason as to why we hold public debates. Not to show two sides of the argument, not to cross bridges, not to explore ideas, but to persuade. This can be done by the merit of your arguments or the appeal of your character, it matters not, since the reason is the same.
On a quick side note, if we indeed wanted to come up with solutions to an issue, this is best done in private, where all people (including us) are comfortable enough to share their real concerns and arguments. This, however is much harder to do, than to just argue against something.
The second and more important reason from a selfish perspective, is the estimating of the value of our ideas. We usually find ourselves convinced of a particular worldview, and this is quite self evident in our heads. We think that we have solid arguments for our views, and we think that should any counters are encountered, that we will leniently subdue them. The usual example is when someone argues to themselves as what they would say in a given situation. This notion that we can come up with a polished reply in a pressured environment is usually fallacious, but the larger delusion is that this reply is the “final word” and that our adversary does not have an appropriate answer to our counter. Thus, many of us find ourselves surprised when presented with an opposing worldview, and shocked when this conflicting argument seems to resonate logically in a hurtful way. I purposely said shocked, since the usual response to a strong counter argument is strong anger. People tend to get emotional over their views and stick to their emotions rather to the pursuit of truth. This is the reason we should indulge in public debate, since if the opponent isn’t able to convince us that there might be a problem in our point of view, than the feedback from the public will indeed force us to either rethink our stance or to rethink the persuasive value of our arguments.
Thus, if we want to estimate the weight of our opinions, and see if they can hold up anywhere else but in our head, we must debate them – publicly.