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This says that $\mathcal{O}_{\mathrm{Spec}(A)}$ is the universal localization of $A$, that is the universal (initial) way to turn $A$ into a local ring. (In contrast, if you restrict your search to ordinary rings, i.e. sheaves of rings on the one-point space, then this optimization problem only has a solution if $A$ possesses exactly one prime ideal. See this MO answer by Peter Arndt for more on this point of view.)

This says that $\mathcal{O}_{\mathrm{Spec}(A)}$ is the universal localization of $A$, that is the universal (initial) way to turn $A$ into a local ring. (In contrast, if you restrict your search to ordinary rings, i.e. sheaves of rings on the one-point space, then this optimization problem only has a solution if $A$ possesses exactly one prime ideal. See this MO answer by Peter Arndt for more on this point of view.)

To go full circle, a variant of this construction is to not give the frame of opens explicitly, but construct it as the Lindenbaum algebra of a certain propositional geometric theory. For any such theory, there is a locale whose points are precisely the models of the theory in $\mathrm{Set}$. In our context, we use the geometric theory of a filter in $A$. Its Lindenbaum algebra is then a locale (verifying the right universal property for the spectrum, even constructively) whose points are the filters in $A$ – so classically, its points are in one-to-one correspondence with the prime ideals of $A$. (See this math.SE answer for more details.)

To go full circle, a variant of this construction is to not give the frame of opens explicitly, but construct it as the Lindenbaum algebra of a certain propositional geometric theory. For any such theory, there is a locale whose points are precisely the models of the theory in $\mathrm{Set}$. In our context, we use the geometric theory of a filter in $A$. Its Lindenbaum algebra is then a locale (verifying the right universal property for the spectrum, even constructively) whose points are the filters in $A$ – so classically, its points are in one-to-one correspondence with the prime ideals of $A$. (See this math.SE answer for more details.)

[Note for the curious: The locale of prime ideals yields the spectrum equipped with the flat topology. The locale of detachable prime ideals (or detachable filters, doesn't matter) yields the spectrum equipped with the constructibly topology (sometimes called "patch topology"). In constructive mathematics, a subset $U$ of a set $X$ is detachable if and only if for any $x \in X$ either $x \in U$ or $x \not\in U$.]

[Note for the curious: The locale of prime ideals yields the spectrum equipped with the flat topology. The locale of detachable prime ideals (or detachable filters, doesn't matter) yields the spectrum equipped with the constructible topology (sometimes called "patch topology"). In constructive mathematics, a subset $U$ of a set $X$ is detachable if and only if for any $x \in X$ either $x \in U$ or $x \not\in U$.]