According to this post Intuition for group homology, I wonder what is the intuition for Hochschild homology.

The Hochschild homology is defined as the homology of this complex chain. Given a ring $A$ and a bimodule $M$. Define an $A$-module by setting \begin{equation} \displaystyle \notag C_{n}(A,M)=M\otimes_{k}A^{\otimes n} \end{equation} and for each $n$ the maps $d_{i}:C_{n}(A,M)\rightarrow C_{n-1}(A,M)$ as follows. Define \begin{equation} \displaystyle d_{0}(m\otimes a_{1}\otimes\cdots\otimes a_{n})=ma_{1}\otimes a_{2}\otimes\cdots\otimes a_{n} \end{equation} and \begin{equation} \displaystyle d_{i}(m\otimes a_{1}\otimes\cdots\otimes a_{n})=m\otimes a_{1}\otimes\cdots\otimes a_{i}a_{i+1}\otimes\cdots\otimes a_{n},\quad 1\leq i\leq n-1 \end{equation} and \begin{equation} \displaystyle d_{n}(m\otimes a_{1}\otimes \cdots \otimes a_{n})=a_{m}m\otimes a_{1}\otimes\cdots\otimes a_{n-1}. \end{equation} When $i<j$, one can check that $d_{i}d_{j}=d_{j-1}d_{i}$. We define a linear operator \begin{equation} \displaystyle b=\sum_{i=0}^{n}(-1)^{i}d_{i} \end{equation} and an $ A$-module \begin{equation} \displaystyle C_{*}(A,M)=\bigoplus_{n\geq 0}C_{n}(A,M). \end{equation} One can verify that $b:C_{*}(A,M)\rightarrow C_{*}(A,M)$ is a differential. The differential complex $(C_{*}(A,M),b)$ is called a Hochschild complex. The homology theory defined by a Hochschild complex is denoted by $H_{*}(A,M)$ and called a Hochschild homology. If $M=A$, the Hochschild homology is also denoted by $HH_{*}(A)$.