From the source I'm reading (S. C. Coutinho, A Primer of Algebraic D-Modules), $A_n$ is defined to be the subalgebra of $\text{End}(\mathbb{C}[x_1,..., x_n])$ generated by the operator $$x_i : \mathbb{C}[x_1,..., x_n]\to \mathbb{C}[x_1,..., x_n]$$ given by multiplication with $x_i$, i.e., $$x_i(f(x_1,...,x_n))=x_i \cdot f(x_1,....,x_n)$$ and the operator $$\partial_i:\mathbb{C}[x_1,..., x_n]\to \mathbb{C}[x_1,..., x_n]$$ given by $$\partial_i(f(x_1,...,x_n))=\partial f(x_1,...,x_n)/\partial x_i,$$ subject to the relation $$\partial_i x_i-x_i\partial_i=I,$$ where $I$ is the identity operator.
Elsewhere, I have read that this algebra may be thought of as the algebra of linear differential operators with polynomial coefficients.
Questions: (1) How are these two definitions equivalent?
(2) Are we are also defining some kind of action of $\mathbb{C}[x_1,..., x_n]$ on $A_n$, or is there an inclusion because then it would make sense how the coefficients are coming about.