As an example.
Suppose you get an interest in your bodily functions, and try experimenting with foods and stuff. You want to understand this stuff.
At first your mental model will be very confused. You may think that you can sense your (say) sodium levels while ignoring much larger effects you can't name yet, and confuse them with those you have named.
As you "diagonalize" your experiments and results, you find that the one thing that dominate health is the "quantity of inflammation".
inflammated vs non-inflammated can then be seen as an "eigenvalue" of bodily function space. Having this first, dominant eigenvector that you can feel will help you identify the second, then the third... until you get into irrelevancy territory.
This is all assuming linearity of things, so of course it has limitations, but your "System 1" neural processes could also be roughly modelled with dot products, from higher dimensional spaces to a single value (ref: Thinking Fast and Slow).
What I wanted to say is that singing leassons taught me to decipher the non-eigen vectors from those uncorrelated (important factors in an outcome).
What happens, at least in singing, is that this diagonalized learning strongly outperforms whatever I had before. For other domains I may very well trick myself :)
The analogy is a metaphor with linear algebra, in linear algebra / PCA you find eigenvectors to decorrelate things.