Interpreting the new dimensions after PCA

To give meaning to the Axis of the PCA, you can study the scalar product between the two new axes and all your original axes (you have to normalise the vectors before doing that).

Those axes with a high scalar product will be highly associated the with the new axes and viceversa. This may help you interpret them.

You could also check t-SNE which is a dimensionality reduction technique based on the probability distribution

One way to explore the mapping between the original dimensions and and PCA dimensions is to look at something called the factor loadings. These are essentially projections of your original dimensions into your PCA space. From this, you can see which of your original features are aligned with your new dimensions, or are aligned with one another.

An example of how to generate a PCA plot with factor loadings in R can be found here, to generate a plot like the one shown below:

Here, we can see that the PC1 axis is aligned with the Petal Length and Width, indicating that higher PC1 value is strongly associated with longer/wider petals. Sepal length is also in a similar direction, so PC1 captures a good bit of Sepal length variability as well. Sepal width, on the other hand, is related to both PC1 and PC2.