This question has been bugging me for a while and I can't seem to make sense of it on a clear conceptual level.
The theory of local rings is given by taking the theory of rings and adding the axioms
\begin{eqnarray}
(0=1) \vdash \bot \\
x + y = 1 \vdash \exists z : (xz = 1) \vee \exists z : (yz = 1)
\end{eqnarray}
This theory is geometric and thus has a classifying topos $S[\text{Local Rings}]$. With a bit of work one can show that this topos is actually given by the Zariski topos, i.e. given as the topos of Sheaves on the Zariski site $CRing_{fp}^{op}$ of finitely presented rings with coverings given by partitions of unity.
As an algebraic geometer with an acquaintance of the functor of points POV this is already a nice fact, as this basically means that scheme theory is a straight up consequence of the notion of a local ring (schemes can be identified with those sheaves that can be covered by representables).
But this is not where the story ends. Write $S[\text{Rings}]$ for the classifying topos of the theory of rings. A ring $R$ is equivalent to a geometric morphism $\text{Sets} \rightarrow S[\text{Rings}]$ (that's just how classifying toposes are defined after all). We also have a geometric morphism $S[\text{Local Rings}] \rightarrow S[\text{Rings}]$. It turns out that the topos theoretic pullback along those two geometric morphism is just the topos of Sheaves on $\text{Spec}(R)$. (This was mentioned in this mathoverflow post by Peter Arndt) (EDIT: As was pointed out by Simon Henry below, this should be taken with a grain of salt until it is clear in what sense this pullback is meant.)
$\require{AMScd}$
\begin{CD}
Sh(Spec(R)) @>>> \text{Sets}\\
@V V V @VV V\\
S[\text{Local Rings}] @>>> S[\text{Rings}]
\end{CD}
hence the construction of the spectrum of a ring follows directly from looking at the fibers of the geometric morphism $S[\text{Local Rings}] \rightarrow S[\text{Rings}]$. The slogan is "the spectrum of a ring is the universal way of making the ring into a local ring" (the corresponding local ring is given by the structure sheaf, which is a local ring internal to $Sh(Spec(R))$ and given by the geometric morphism $Sh(Spec(R)) \rightarrow S[\text{Local Rings}]$).
Moreover, locally ringed spaces as special cases of locally ringed topoi are considered by some writers as the right notion a "geometric" space – for example Jacob Lurie generalizes basic notions of geometry to higher geometry using locally ringed spaces. There's a natural notion of what it means to be locally isomorphic to a given locally ringed space and the concept of smooth manifold, complex analytic manifold and also scheme derive naturally from that. The category of locally ringed topoi is simply the overcategory (or slice category) of $\text{Topos}$ over $S[\text{Local Rings}]$ – a nice fact is that this automatically gives the right notion of a smooth map between smooth manifolds, holomorphic map between complex analytic manifolds, etc.
Some other remarks from the side of constructivist mathematics are that linear algebra in a constructive setting works best over local rings (as there are different notions of field in constructive mathematics).
So to come to my question: What is going on here? Let me be specific: I can grasp the concept of a group as a syntactic description of what we mean by the symmetries of an object – what is the underlying essence of a local ring? What part of our human intuition do they describe naturally?
A little remark: There is a partial answer to be found in the following: A local ring has a canonical apartness relation $x \# y$, where $x \# y$ iff $x-y$ is invertible. In fact a ring object in the category of sets equipped with apartness relations is the same thing as a local ring. I am however not yet satisfied with this.
Best Answer
I'm not sure if this constitutes a full answer, and a lot of it has already been said in some form by HeinrichD in the comments. Because your question is ultimately one of philosophy, I will focus mostly on history and philosophy (and a few applications); not so much the categorical or logical interpretations.
History. In functional analysis, the Gelfand representation shows that locally compact Hausdorff spaces can be thought of as commutative C*-algebras (but it's not quite an equivalence of categories). Then Grothendieck came around and reversed this philosophy: we can take any commutative ring $R$, and define a space $X$ such that functions on $X$ are by definition given by $R$.
Philosophy. So in my view, the philosophy of commutative rings is that they behave like functions on a space, with the operations on functions that we are used to (notably: multiplication and addition).
Local rings add one thing to this profile: a notion of vanishing or nonvanishing at a point of a given function (depending on whether or not it is in the maximal ideal). Perhaps we should think about this in light of the Stone–Weierstraß theorem: a 'good' notion of function should be able to separate points, and for this you need a notion of vanishing at a point. Note that for a general ring, the statement $p(x) \neq p(y)$ does not make sense, because $p(x)$ and $p(y)$ take values in the rings $\kappa(x)$ and $\kappa(y)$ respectively. However, the statement "$p$ vanishes at $x$ and not at $y$" does make sense.
Can we do with less? Yes, we can. For example, replacing commutative rings with pointed monoids (the point corresponding to $0$ in a ring) gives another geometric theory, for which some people suggestively use the word $\mathbb F_1$-schemes.
In terms of the philosophy above, we do away with addition, but we keep the notions of multiplication and of identically vanishing of functions. I think it is at this point that our geometric intuition leaves us behind, and perhaps this is the whole point of your question...
A general theory? It might be possible to abstract away what properties of local rings give us the most general setting in which one can do algebraic geometry. But it's not clear at all that there is an answer to this question, for it depends heavily on what properties you want your geometric theory to satisfy.
Perhaps the only way to get started on answering this question is by examining the many different 'generalised algebraic geometric' notions that have been defined and used, and carefully studying their properties. There are many different generalisations that people use, and I think a systematic study is neither possible nor desirable: it depends strongly on the application one has in mind.
Some examples. Here are some generalised/altered notions that people use:
All of these (and many more) notions are being used by people to actually prove things that are formulated extrinsically (without reference to the newly developed theory). These theories all share properties in common with the theory of schemes, but their geometric behaviour is very different each time, depending on the desired application.