This is related to Weibel Prop. 4.4.11 of Homological Algebra. Induction step is clear to me but I do not find the base case clear to me. $\operatorname{pd}(A)$ denotes projective dimension of $A$.(i.e. shortest length of projective resolution.)
$R$ is local ring with residue field $k$ and $A$ is f.g. over $R$. Then $\operatorname{pd}(A)=0$ iff $\operatorname{Tor}_1^R(A,k)=0$. (This is already phrased in base case.)
Original statement: Prop 4.4.11 If $R$ is local ring with residue field $k$, then for every f.g. $A$ and every integer $d$, $\operatorname{pd}(A)\leq d$ iff $\operatorname{Tor}^1_{d+1}(A,k)=0$.
Since $R$ is local, take minimal number of generators of $A$ by Nakayama's lemma. Say there are $n$ of them. So I have a short exact sequence $0\to K\to R^n\to A\to 0$ with $R^n\to A$ canonical projection map and $K$ kernel.
Since $\operatorname{Tor}^R_1(A,k)=0$, this implies $0\to K/mK\to k^n\to A/mA\to 0$ is exact. Furthermore, $K/mK=0$.
$\textbf{Q:}$ The book says $K$ is finitely generated. Why so? If $R$ is noetherian, then it is possible to conclude $K$ f.g. and it follows from Nakayama that $K=0$ which implies $A$ free and hence projective. However, the statement is made for local ring $R$ without Noetherian hypothesis.