I don't know if it was me or my lack of sleep, but several times in this reading I got lost. However, I do think that I got the overall point of it.
Lewis created his own word "Bulverism" for the common way that people at his time, and still today, argue against other people's points. He says that everyone assumes that the other person is wrong before addressing the why. In other words, the person is fighting for a victory, and not for the real truth. But in reality what we need to do is first show that they are indeed wrong, and then start to explain why. In this way we will be no longer be blinded by our fast need to win, and we can work together towards a real truth.
I think, and it was also discussed in class, that this points to humility. Humans need to be humble in order to realize that they might be wrong, this way people will not assume, and will not Bulverize.