The common wisdom is that when creating a product you must solve a problem or need people have. For example, Facebook solves the social problem, and Evernote is your "second brain".
However, it seems that the luxury industry doesn't solve problems.
They do solve a problem. They may just not always be problems that we can relate to. Keep in mind that we all have different needs/problems. And emotion can play a part in the kind of problems we have and how we solve those problems.
As to why luxury items can be sold for high dollar amounts: because people will pay for them. Some people have more money than they can spend in a lifetime, so why not splurge on high end items. Others may feel pressured into buying expensive items if those around them do so. The human mind is complex, and there are many reasons why people buy luxury items. That's why it pays to understand your market and their needs. If you can offer them what they want, you can charge a premium for it. Take for example designer clothes. A simple problem like "I need something to wear to the party next week" can be solved in many ways. It's up to you to know your audience and get inside that person's head to offer them the right solution.
They are solving the problem that there is a demand for products of higher quality and for ways to demonstrate higher social status.
These products usually have a large emotional value attached to it that can't exactly be rationally justified.
It's not always a 'problem' and a solution. Sometimes it's a better/new service that answering a need you didn't had.
It could be: faster, cheaper and/or better quality. In the 'luxury' space there are many successful startup that do one of the things I've mention. For example:
and many more... Airbnb.com on their 'top' section of places also give you the option to get great quality for less.
People (that can pay) wish to get quality and thus there is a room to innovate new services and product that provide it.
I think you hit on the answer in your question when you mentioned "or need people have". Many people feel a need to purchase expensive things. IMO the more outrageous the price the more effective the luxury product is.
Maslow's hierarchy of needs is one way to answer your question. Needs are of different kinds and degrees. According to Maslow they are progressive in nature in the order of physical needs, security needs, social needs, esteem needs, self-actualization needs. When you have loads of money you most likely have been covered for the first three. hence, your mind tends to satisfy the esteem needs by buying stuff that creates news and enormous esteem value for you. There comes a point sometimes where you tend to get weary of it and seek for something deeper (self-actualization needs).
That is the definition of a niche markets - a group that has a specific desire / trait.
It doesn't mean that it is universally applicable. If you attempted to explain the value proposition applicable to a World of Warcraft group to your grandmother, I doubt she would get it either.
Regarding the "social problem" that facebook addresses, I would point to this toyota commercial http://www.youtube.com/watch?v=TUGmcb3mhLM.
Facebook probably created more problems than solving any problem - but that doesn't mean that it didn't have something to offer its target market. The initial target market (college students) wanting to learn about (and status of) their classmates is real important to them. And it did scale beyond that initial proposition (everywhere except the financial markets, it seems).