Maybe the following can help motivate this definition. In the case of $\mathbb{R}$-valued functions on a topological space $X$, if $\mathcal{U}$ is a covering sieve on $X$, so that the union of its open sets is $X$, then defining a function $\mathcal{U}$-locally on X is equivalent to defining a continuous function on $X$ (SGA 4 1/2, I, 2.3). One also defines a notion of vector bundles given $\mathcal{U}$-locally, and in this case the corresponding statement is that if $\mathcal{U}$ is a covering sieve, then there is an equivalence between the categories of vector bundles on $X$ and vector bundles given $\mathcal{U}$-locally (SGA 4 1/2, I, 3.3). And finally in the case of (flat) schemes over $X$, the corresponding statement is that if $\mathcal{U}$ is the sieve generated by a family $\{U_i\}$ of flat schemes over $X$, and the union of the images of $U_i$ is $X$, then there is an equivalence between the categories of quasi-coherent modules on $X$ and quasi-coherent modules given $\mathcal{U}$-locally (SGA 4 1/2, I, 4.5).