Let $R$ be a commutative ring. Any $R$-module has a presentation $R^{(J)}\rightarrow R^{(I)}\rightarrow M\rightarrow 0$. The associated module functor $M\mapsto \tilde M$ is exact and so preserves exact sequences, meaning every element of its essential image comes with an exact sequence $\mathcal O_X^{(J)}\rightarrow O_X^{(I)}\rightarrow \tilde M\rightarrow 0$. This is equivalent to an exact sequence at the level of stalks.
What I don't understand, is why it implies that each point $x\in X$ has an open neighborhood $U$ such that we have an exact sequence $$\mathcal O_X^{(J)}\mid_U\rightarrow O_X^{(I)}\mid_U\rightarrow \tilde M\mid_U \rightarrow 0$$ with $I,J$ varying with $x$. I think this has to do with basic sheaf theory but I'm not sure what I'm missing.