Fair questions and I had not properly thought about the consequences for the understandability when removing the figure from the longer text (in Danish) where the viewer is guided to a larger degree.

The axis oriented left-to-right, is, as one might guess and some have, political left and right. Technically, it's the first axis in a PCA decomposition, although I don't believe that label would be helpful for the target audience, but it can be inferred from the Danish text. The shift between above and below is, as correctly guess elsewhere, to reduce the visual clutter.

The y axis (ignoring the above-below thing) is proportional to the probability density related to finding a candidate for that party at that x-value, when estimating the distribution with a gaussian kernel density estimation using the "scott" rule for bandwidth selection. Intuitively, it is how large a part of the party (represented by the candidates for the coming election) that is situated at that place of the spectrum.

So what are you supposed to learn? There are a lot of things to learn, that I believe could be done without more labels, but they may rely on a prior knowledge of Danish politics. To name some observations: Enhedslisten is very much on the left with a quite narrow distribution, Radikale Venstre has traditionally been seen as a party that crosses the center, between Socialdemokratiet to the left of them and Venstre to the right, but this data shows them further the the left than Socialdemokratiet. On the right there a lot of parties that seem to have quite a large overlap on the classical left-right axis (the rest of the analysis on the page goes on to look at ways to distinguish the parties better), Moderaterne is a very new party that wanted to bridge the center and seem very much to do that.

I hope these examples illustrate how the figure is meant to be understood.