I am trying to understand the theory of simplicial commutative rings or animated rings. I just find a remark in Lurie's book Spectral Algebraic Geometry:

Remark 25.3.1.7. Let $f : R[x_1,\ldots ,x_n]\to R[y_1,\ldots,y_m]$ be a homomorphism of polynomial rings over $R$, given by $x_i\mapsto f_i(y_1,\ldots,y_m)$. For every object $\DeclareMathOperator{\calg}{CAlg^\Delta_\textit{k}} A\in \calg$, composition with $f$ induces a map of spaces $$\DeclareMathOperator{\mapcalg}{Map_{CAlg^\Delta_\textit{R}}} \mapcalg (R[y_1,\ldots,y_m],A) \to \mapcalg R([x_1,\ldots,x_n],A). $$ Passing to homotopy groups at some point $\eta\in\mapcalg R([y_1,\ldots,y_m],A)$, we get a map $\pi_\ast(A)^m\to\pi_\ast (A)^n$. For $\ast =0$, this map is given by $$ (a_1,\ldots,a_m)\mapsto (f_1(a_1,\ldots,a_m),\ldots,f_n(a_1,\ldots,a_m)). $$ For $\ast >0$, it is given instead by the action of the Jacobian matrix $[\frac{\partial f_i}{\partial y_j} ]$ (which we regard as a matrix taking values in $\pi_0(A)$ using the morphism $\eta$).

I do not understand why the induced map of higher homotopy groups is given by the Jacobian. Also I cannot see how the map of polynomial rings influences the homotopy groups. It seems too mysterious for me.