Bismillahirohmanirohim (in the name of Allah the Most Merciful, the Most Compassionate)
A brief explanation on why we should not boast and be proud is because, a proud person will always think that what he has is good (regardless of material or intellectual capabilities), and what others have is bad. Therefore, when a person is proud of his work usually they do not strive to improve on it and also since they think what others have is bad, it means that indirectly they are looking down upon others.
I've read some interesting stuff on this "proud" observation from Ibn Hazm. He went to a person and asked "Why are you proud?" That person replied, " I am nobody's slave!". Then Ibn Hazm pointed out that most of the people are not slaves but they are not proud or boastful compared to that particular person. This "proud" is considered as a "false pride". It is a trick being applied into the human mind. In my opinion of today's context of "false pride" is that we are made to feel proud of owning certain things where else it is those certain things who are proud to own us.
Most of us today own certain items just for the sake of the "brand" and not because of its capabilities. For example, Macbook is considered desirable to most of us even though most of us only use it for casual use. Why not use the pc or laptop? It is understandable if a person really needs the technology available in the mac book owns a Macbook because its technology is not available in any ordinary laptop. But why own a Mac if the technology is rarely being used? This is what it is meant by "false pride". We feel "happy/excited/proud" of owning such things but in reality it is our mind being played by tricks.
A person who likes to own these "brands" will usually not be satisfied or happy. There will be a time when that person will not feel the item/material is special anymore due to others owning the same item. There will also be a time when there is a new/better item and it will cause that person to be dissatisfied or unhappy. These feeling of dissatisfied/unhappy will then recur up to a point that it becomes a mental illness and will be a chain reaction towards one's life and others. This will become an unachieveable satisfaction and will only rely on materials/items for temporary happiness. Therefore it is important that we recognised that Allah is the only One who can make us feel happy and it is better for us to strive to enter paradise and not strive for worldy things.
When a person owns an expensive thing/brand, jealousy is the nature that would normally come from some of those who cannot afford it. Hence, the need to get a fast solution to own those things. This is when some will be tempted to do undesirable things like stealing and etc. So, it is important not only for the person who is a "show off" to be moderate but also for the best of the community.
We should try and change our mentality or perception to be satisfied even without the "best" of wordly things, then it will be easy for us to be happy with our lives growing a healthier community. Let this be a reminder to myself and others not to be arrogant and boastful.
Surah Luqman verse 18: "Do not turn your face away from people, nor tread haughtily upon earth. Allah does not love the arrogant and the vainglorious." (19) "Be moderate in your stride and lower your voice. Verily the most disgusting of all voices is the braying of the donkey."
Surah al-nisa verse 36: "Serve Allah and ascribe no pratner to Him. Do good to your parents, to near of kin, to orphans, and to the needy, and to the neighbour who is of kin and to the neighbour who is a stranger, and to the companion by your side, and to the wayfarer, and to those whom your right hand possess. Allah does not love the arrogant and boastful"
In my opinion, these verses of arrogant/vainglorious/boastful are interconnected with the community around one. It is obvious that one can act arrogant and boastful only when others are around and the reminder of these verses are to do good towards people around you no matter of one's status/rank.