Why is $P(a \text{ and } b)$ maximized when $P(a \text{ or } b)$ is minimized?

$A \cap B$ is the overlapping part in $A \cup B$.

Imagine you have two sheets of paper and you want to maximize their surface.

If they overlap, you can increase the surface by making them overlap less.

In the relation $$x=y-z,$$ you get the largest value for $x$ when the value of $z$ is a minimum. That is, you would get the largest difference between two numbers if the number you are subtracting is made as small as possible.