For an associative algebra $A$ one can define the Hochschild cohomology of $A$ as $ HH^n(A,A):= Hom_{\mathcal{D}(A^{op} \otimes A)}(A, [n]A)$ (this definition also works for the graded and dg cases as well). There is a lot of information contained in this gadget: when $ n=0$ it is the center and when $n=1$ it is the lie algebra $\text{Der}(A)/\text{Inn}(A)$ and when $n=2$ it says something about the deformation theory of the algebra. Also there is a cup product and Gerstenhaber bracket on $HH^*(A,A)$.

My question is: What is the reason/motivation for the definition $ HH^n(A,A):= Hom_{\mathcal{D}(A^{op} \otimes A)}(A, [n]A)$, if you didnt know about all this higher structure what would lead you to study it in the first place?