What does a higher cost of living mean to a city?

it means everything is more expensive - rent, food, entertainment.

it is usually caused by people in the city earning a lot of money. Which in turn does mean that there are more and better things available in the city. But you cannot buy them unless you earn a lot of money too.

Not much. All cities have a high cost of living just because they are cities.

I suppose you can argue that it indicates just how desirable the city is to live in. But even there, it isn't just the cost but also the expected income. Most cities have median incomes that are higher than the country as a whole, but generally not as much higher as the cost of living. So the real metric for attractiveness would be (cost of living / median income).

The primary element in cost of living is the cost of real estate: housing for the individual and rents for all the retail establishments. The more people want to live in the city, the higher is the cost of real estate, rents, etc. The more space constrained a city is (think of Manhattan on its island, San Francisco on its peninsula, Vancouver squeezed between the mountains and the coast, etc.), the higher the real estate costs and hence the cost of living for a given number of people.

In the end it balances. People choose to go to a city and to which city based on costs vs. benefits. Higher pay is one benefit, but there are others: job opportunities, opportunities to meet niche groups of people, etc.
http://www.creativeclass.com/_v3/whos_your_city/overview/