Before I answer this question, let me say a bit about how I understand it. I do not think you can compare the US minimum wage with this article. There would be very little to say other than that the US minimum wage is much lower than the wage that fast food restaurants are required to pay in Denmark. What I think you are asking is how we can reconcile Denmark’s minimum wage (it is not technically a minimum wage because it is not imposed by the government, but it is still a price floor) with the conventional wisdom in economics on the subject of minimum wages. I will base my answer on this understanding.
In the academic field of economics, minimum wages are seen as a bad thing. This is because economists believe that they create a surplus of labor. When the government sets a minimum wage that is too high, more people want to work. The quantity of labor that is supplied goes up. However, at the same time, the quantity of labor that is demanded goes down. This is because employers cannot or do not want to buy as many hours of labor now that the price of that labor has been artificially raised. Therefore, we get a surplus. We get more people wanting to work and fewer employers wanting to hire. To this way of thinking, minimum wages hurt workers because they cause employers to hire fewer workers.
The question, then, is how Denmark can get away with imposing a $20 per hour minimum wage on its fast food restaurants. There are several points to be made on this question. First, Denmark has a high cost of living and high taxes. This means that its workers need to make more money in order to be able to survive. Second, the higher wages in Danish fast food restaurants allow those restaurants to keep their employees for longer. This saves them money on training new employees and it means their employees will be more experienced and therefore more efficient at their jobs. This helps make them more valuable to their employers than US fast food workers typically are.
The biggest factor, however, is the difference in cultural expectations between the United States and Denmark. American business owners expect to get higher profits and American consumers expect lower prices than their Danish counterparts do. The article points out that Danes expect their burgers to be somewhat expensive. It quotes a Danish expert who says
We Danes accept that a burger is expensive, but we also know that working conditions and wages are decent when we eat that burger.
American consumers are less likely to accept expensive burgers. They (we believe) want their fast food to be as cheap as possible. At the same time, American business owners want more profit. The Danes are willing to take lower profits, the article says, because they value economic equality. The article quotes a Danish businessman saying
We don’t want there to be a big difference between the richest and poorest, because poor people would just get really poor. We don’t want people living on the streets. If that happens, we consider that we as a society have failed.
So, there are a number of reasons why Danish fast food restaurants can pay such high wages, but the most important is the difference in cultural assumptions. This difference makes it so that Danish employers will tolerate paying higher wages and Danish consumers will tolerate paying higher prices. Therefore, the demand for labor does not drop as much as we might expect when the minimum wage is high in Denmark.