# Difficulty interpreting/understanding canonical correlation analysis

I have read many posts on the topic (e.g. this or this) and have gone through a few introductions/tutorials, however, while I do understand the mathematical description, I still have a lot of difficulty understanding the results of CCA on an intuitive level.

Basically, assuming the notation

$$X \in \mathbb{R}^{n\times d}, Y \in \mathbb{R}^{n\times p}$$

for two different data/features matrices, I understand that the goal of CCA is to find the transformations $$\textbf{a} \in \mathbb{R}^d, \textbf{b} \in \mathbb{R}^p$$ such that $$X\textbf{a}$$ and $$Y\textbf{b}$$ are maximally correlated. However, once $$\textbf{a}$$ and $$\textbf{b}$$ are found, I fail to understand what the information that they reveal about the original feature set is (and therefore, I think that I'm missing the motivation of CCA). For example, consider a toy example such as the one illustrated by the figure,

where, keeping the definitions for $$X$$, $$Y$$ introduced earlier and using Matlab-style notation,

# we generate 50 examples, each with 4 features that we devide into
# two groups, i.e. p=2 and q=2 in the above.
N=50
X=zeros(N,2);
X(:,1)=linspace(0,2*pi,N).+randn(N,1)'*0.5;
X(:,2)=10*sin(X(:,1));

Y=zeros(N,2);
Y(:,1)=linspace(0,2*pi,N).+randn(N,1)'*0.5;
Y(:,2)=10*log2(Y(:,1)+1);


The last figure is the result of plotting the first canonical variables, and as expected, we see that the resulting variables have a correlation that is close to $$1$$.

All of this is great, but... Well... Now what? In particular:

1. What does that tell about the original set of variables/features/descriptors (i.e. the columns of $$X$$ and $$Y$$)?

2. What is the benefit of working with the basis that give us a diagonal correlation matrix?

3. What if the first canonical variables had low correlation? What would be the interpretation of that?

PS: I mostly work in computer vision, so please excuse me for the inevitable misuse of jargon and approximations.