Maybe it's worth spelling out the easy case of projective space (and Veronese embedding).
Take the standard exceptional collection $<O,O(1),...,O(n)>.$

The derived category of a hypersurface X of degree d has then a semi-orthogonal decomposition $<A_X, O, O(1), ..., O(n-d)>$. We might think of $A_X$ as being the "interesting" part of the derived category of X.
We can also intersect further and $D(X_1 \cap X_2) = <A_{X_1 \cap X_2}, O,\ldots, O(n-2d)>$.

Take the universal hypersurface of degree d, call it $H$.
This sits inside $P^n \times P^N$, where $N$ is the ambient dimension of the degree d Veronese embedding.

The category $H$ (being itself a hypersurface) has a decomposition where we call the interesting part just $A$.

For each point $p \in P^N$ we have a hypersurface $H_p$ and thus an interesting category $A_{H_p}$.

The "homological dual" variety is a $Y$, mapping to the dual $P^N$, such that $D(Y)$ is the total $A$. In some sense, $Y$ is parameterezing all the categories $A_{H_p}$.

The point of HPD is then the compatibility between passing to base loci and base changing Y.

If $L < P^N$ is some linear system, then the derived category of the corresponding base locus has a decomposition and the interesting part is the interesting part of $D(Y \times_{P^N} L$.

(this is perhaps too sloppy to be useful but I tried)

The Viennese school of Katzarkov, Ballard, Favero etc. has shown that in this context (of degree d Veronese embedding) there is a (sort of tautological) dual. Unfortunately this dual is a highly non-commutative variety, in fact it's the dual $P^N$ with a sheaf of $A_\infty$-algebras over it.