I'll toss my two cents into the discussion, although remember that two cents are worth almost nothing these days.
Denmark is a small country with few of the social challenges or economic obligations facing larger countries such as the United States.
Denmark is a small country with a relatively miniscule defense budget and no major defense obligations. If Denmark were attacked by a larger country (say, Germany), the Danes could not easily resist. They benefit from membership in NATO but do not contribute nearly as much to the organization as larger countries do, either financially or in raw manpower.
Interestingly, Denmark was a founding member of the European Community but is not a member of the Eurozone. If it were a member of the Eurozone, it might be expected to help prevent the bunkruptcy of such other Eurozone member nations as Greece, Portugal, Spain, Italy, Ireland, and perhaps even France, to mention just a few. It is precisely because Germany is expected to contribute to such causes that many Germans are now very skeptical about the Eurozone.
The entire nature of your question (as well as the poll you are referring to, which I am familiar with, by the way) should be considered, at best, questionable. Many of the previous posters have echoed this.
What is happiness? A blind feeling of content? Knowledge of faith and the afterlife? Knowing that one is going to be taken care of no matter what happens? A feeling of trust in government and community? It's a different answer for every single person!
My original thoughts were stolen by akannan. It's a poll designed to show that people living in a liberal society are happier than ones living in a conservative society designed to highlight the general differences between capitalism and socialism.
Some of the factors are more prevalent in Denmark and less prevalent in other places. For instance, their incomes are comparable across the country so that a garbage collector lives in the same kind of neighborhood as a doctor and earns a comparable amount. Prestige is equally comparable: the garbage collector gets the same kind of respect as the doctor. Children are safe within their families and society (NOT true for children in America ... uh oh ... controversy): baby prams are left unattended (as is also seen in Switzerland); bicycles are left unlocked; trust in other people and government are both high. Education is available to all with equity and with ease--higher education doesn't come with an enormous student loan price tag that requires trading off financial ease for knowledge and expertise.
Are you kidding me? Have you never had a Danish pastry? Those things are DELICIOUS! No, seriously, studies of the happiness of nations are always fraught with difficulties in terms of how on earth do you quantify such a nebulous term as happiness. As other editors above comment, you can't equate happiness with material wealth, and some would argue the richer you get the unhappier you are. In addition, I have met many people who are incredibly poor who are happier than many rich people I know. So, let's take the results of such surveys with a pinch of salt.
Happiness is a cultural phenomenon. Americans are miserable by nature. We always want more. We are never satisfied no matter how much money, power or health we have. The Danes, on the other hand, focus on happiness and take pride in it. They are happy because they want to be happy.
There is no simply way to answer this for many reasons. How does one measure happiness? What is happiness anyway? To answer these questions and to set a standard is important. These are crucial methodological controls that need to be established. With that said, perhaps the best way to approach this issue is to ask Danes why they are happy. If we do this, they usually give two reasons. First, they point out that most of the society is not made for the upper class. Just the opposite, nearly all things are catered to the middle class. Hence, there is a sense of contentment, which is key. There is little of the mentality of "keeping up with the Jones." Second, they mention the great services that the state provides. This comes at a price though - extremely high taxes. This fact also has another benefit. People tend to decide on an occupation based on what they like and not based on earning potential. This creates happiness as well.
If we put these things together, then there really is a sense of contentment.
Of course, we cannot know why the Danes are happier (according to some measures) than other people of the world.
To liberals, the Danes' happiness can be attributed to their expansive welfare state. Liberals tend to argue that the Danes are happy because their lives are relatively secure and because they have a good balance (enforced by the government) between work and home life. Danes, for example, get a long period of paid parental leave when they have children. After that, they are able to send their children to day care facilities at very low cost to themselves.
Liberals argue that these kinds of benefits make Danes feel secure and fulfilled. This leads to them being the happiest people in the world.