Here's my intuition on how you could have invented the Hausdorff distance. Hopefully it helps.
You want a metric that tells you how far two compact sets are from being the same. And since these sets happen to be subsets of a metric space, you ought to define your metric in terms of the distances between the points of $A$ and $B$.
Suppose $A\ne B$. Then either there is a point in $A$ that is not in $B$, or there is a point in $B$ that is not in $A$ (or both).
Let's say there is an $a\in A$ with $a\not\in B$. How far is $a$ from being in $B$? Well, the least you have to move $a$ to get it into $B$ is the distance to the closest point in $B$, which is $\inf_{b\in B} d(a,b)$. So that's the distance from $a$ to $B$, which we might as well call $d(a,B)$. Observe that if $a\in B$ then $d(a,B)=0$, and because $B$ is compact, if $a\not\in B$ then $d(a,B)>0$.
Now there are lots of points $a\in A$, some of which may be in $B$, and some may not. As long as there exists any $a\not\in B$, that is, any $a$ such that $d(a,B)>0$, we want to know about it. So we ought to take the supremum: $d_1(A,B)=\sup_{a\in A}d(a,B)$. This also means that every point in $A$ is at most $d_1(A,B)$ away from $B$.
Finally, $d_1(A,B)$ only tells us if there is a point in $A$ that is far from $B$. We want the Hausdorff distance $d_H(A,B)$ to be large if either there is a point in $A$ far from $B$, or there is a point in $B$ far from $A$. So we define it to be the maximum of both $d_1(A,B)$ and $d_1(B,A)$. And we're done.