Being arrogant is for sure a no-good act but we sometimes do this wrong thing, no matter what our age is. Kids and adults often act and say that they are much better than others. The speech person that day told us about it. He exampled a kid who was arguing with his counterpart and he says that his dad has a gun, his counterpart does the same. He even told that kid (first) that his father does have a gun and it is bigger that his counterpart's father's. You may have experienced this in your life, I have. But the worst is that, we mature people sometimes do the same. We kind of proud of our parent's having or something. We feel proud of having of our partner's having or something. But indeed that thing is nothing, we are like children then. In short, the speech was about takabur or boast that we must actually leave away.
There are several things that can make us be takabur person. First is our knowledge of something. When we thing we know many things, we tend to say them to other persons. It is good to share, but we need to make sure that we do that mere for the sake of of pure good, Allah's sake. It is hard as it is the matter of heart. But keep straightening our intention. The second is our worship. When we think we are better in particular worship (such as reading Quran), there is also possibly for us to act arrogantly to others. But, it doesn't mean that we need not to read Quran or to be smart in it. It is all about straightening the intention as mentioned before.
Hope we're able to maintain this heart, only for Allah..