Look, energy production is hard stuff, and the reporter here clearly didn't understand ANY of the intricacies.
Basically the situation is this: you have a consumption curve that you need to meet at every instance. It is important to understand that this is a curve with daily peaks. These peaks MUST be met or you get riots in the streets.
If you erect a wind turbine it will produce power as the wind blows. Same with solar and the sun. When you match the resulting production curve up against the consumption curve, there will be gaps that you need to fill in some other way.
Nuclear power is a bad way to fill the gaps. Due to high capital costs, to stay economical a nuclear plant usually needs to produce 100% all the time until it needs refueling (which takes a month I think) where it will produce 0%, in other words a flat line with some clearly defined gaps. But we need to match a curve with gaps, so a flat line doesn't help much.
Instead you need something you can dispatch relatively quickly without costs going through the roof. Currently stuff like hydro, biogas, biomass, etc.
In Denmark, besides all the wind turbines we have a bunch of big coal plants. These plants are currently being transitioned to biomass (i.e. wood pills and chips) and will fill in the gaps, as well as produce heat for district heating (which is really big in Denmark, winter's cold up here).
If these plants get into financial trouble, the national grid operator Energinet can increase a fee on each kWh (the PSO) and use the extra income to pay some of the plants for standby services. Besides this, we have really good grid connections to Norway where they have a ton of quickly dispatchable hydro. The connections to Norway are a two-way street - they get cheap wind turbine power in return which makes it easier for them to get through the winter without running out of water (very little water flows to the dams in winter because it's frozen).
Hence, apart from the transportation sector where we're waiting for Tesla and the like to come up with better electric cars, there really isn't anything tricky or hard about the transition away from fossil fuels in Denmark.
It was tricky in the past because wind turbines used to be expensive, but the industry has matured and wind is now the cheapest source of new (undispatchable) kWhs. Really, the only political question left is whether we should try to save some of the biomass by building more off-shore wind turbines.
It's also true that our current path is a bit more expensive than a fossil-based base scenario - I think it's supposed to be around 100-200 USD per inhabitant per year in 2050. So not overwhelmingly expensive.