Why is $mathcal{O}_{X}|_{U_alpha}(D_1 + D_2) cong mathcal{O}_{X|{U_alpha}}(D_1) otimes mathcal{O}_{X|{U_alpha}}(D_2)?$

Given two Weil divisors $D_1$ and $D_2,$ how do we know that $$mathcal{O}_{X}|_{U_alpha}(D_1 + D_2) cong mathcal{O}_{X|{U_alpha}}(D_1) otimes mathcal{O}_{X|{U_alpha}}(D_2)?$$

I’ve seen this fact in a few different places, all given with no explanation, so I assume it’s supposed to be trivial and I just don’t understand divisors well enough.