In Tierney's 1976 paper On the Spectrum of a Ringed Topos (which you can find here) at the top of section 2 we read
Let $A$ be a commutative ring in [a topos] $\mathbf{E}$. We look at the problem of finding the spectrum of $A$ as the problem of finding a free local ring on $A$. That is, we try to find a universal homomorphism $\gamma : A \to L(A)$, where $L(A)$ is a local ring.
Then a page later
That is, we are looking for a right adjoint to the forgetful functor $LR\text{-top} \to R\text{-top}$. When such an adjoint exists, it is characterized up to unique local equivalence, and we call it $\text{Spec}(\mathbf{E},A)$, the spectrum of $(\mathbf{E},A)$.
Here $R\text{-top}$ is the category of topoi with a distinguished ring, and $LR\text{-top}$ is the category of topoi with a distinguished local ring. You can find more details about the exact morphisms in the paper, but they're what you expect.
Tierney then spends some time describing historical constructions, before finally proposing a new construction, introducing a technique where we build a free ring (since this theory is algebraic), then introduce a (grothendieck) topology whose sheaves respect the additional geometric axioms of a local ring. I mention this mainly because I found it instructive, but it's not particularly relevant.
Now, it's clear that this is a good notion of "spectrum", since Tierney proves (on written page 207)
the $\mathbf{E}$-points of $\text{Spec}(\mathbf{E},A)$... correspond $1$-$1$ to the primes of $A$.
(NB: "primes" in this paper are what we would classically call complements of prime ideals. This is to avoid some technical issues with double negation that I don't fully understand)
The question, then:
Obviously this idea of "free" local rings does lead us to a good notion of spectrum. But it's entirely opaque to me why we would consider it at all. Maybe this is my weakness in algebraic geometry showing? Regardless:
What is the motivation for looking at "free" local rings? And what does this have to do with the construction of the spectrum of a ring? I would prefer answers which give concrete examples if possible.
Thanks in advance ^_^