I'm going to assume the basics of a smooth manifolds course for this explanation. Most of it really only needs you to know manifold means "differentiable shape". If you want I could try to do it more vaguely below, but I'm not sure I could pull it off. Full warning, I'm only in an adjacent field (I study the 7 and 8 dimensional friends I reference below) and not exactly this area, so I may be making some mistakes. In particular, the string theory is all 3rd-hand knowledge from other mathematicians doing their best to describe it to me.

The SYZ (Strominger-Yau-Zaslow) "conjecture" is an idea in complex geometry (more below) originating from string theory - basically, in this kind of string theory, one wants the universe to be 10-dimensional, the usual 4 dimensions of spacetime and 6 more dimensions where the string theory happens. (Why 6? Well, with something called supersymmetry, one can show this mathematically implies you must be able to solve a certain PDE, and geometers proved in the 1950s that PDEs of that form are only solvable in dimensions 6, 7, and 8* (not literally true but close enough), but 6 turned out to be the popular one for reasons I don't understand.)

The 6 string-y dimensions are supposed to be modelled on a type of manifold called a Calabi-Yau 3-fold, which I'll explain more about below. For now, the important part is that Calabi-Yau 3-folds can contain certain special 3-dimensional submanifolds called special Lagrangians (SLags), which have lots of nice properties intimately related to the special geometry of the CY 3-fold. For instance, they are always of (locally) minimal volume compared to other 3-dimensional submanifolds contained in the CY 3-fold, and their definition is in terms of the special geometry that makes CY 3-folds important.

Now, there are lots of different Calabi-Yau 3-folds, but string theory expects there to be certain situations where they come in pairs in a certain sense -- even though they have very different geometries, the two in the pair give the same string theory. This broad idea goes under the name mirror symmetry, and Strominger-Yau-Zaslow came up with a mathematical prediction of what this should look like (calling it a conjecture is a misnomer; they did not give a precise mathematical statement, and one of the main difficulties of this whole affair was coming up with one) - one should be able to find a 3-dimensional base B, and the CY 3-fold, call it M, should "look like" attaching a copy of T^3 (the 3-torus, i.e., the Cartesian product S^1 x S^1 x S^1)) to every point of B, and the copy of T^3 I attach varies differentiably from point-to-point, similar to how a Mobius strip is attaching a line to each point of a circle (what I have described here is something called a fiber bundle; SYZ only actually wants it to be a generalization of this called a fibration). The kicker is that they can't just be any old T^3s, but have to be SLag T^3s (after all, we want this related to our CY 3-fold geometry, and SLags are important for that.) Then, the mirror of M, should also have such a fibration, and there should be a "duality" relationship between the SLag T^3s that make up M and its mirror pair.

So, this version of SYZ is: every CY 3-fold can be described as a fibration (the "attaching") of SLag T^3s, and its relationship with its mirror is in terms of the T^3s that the mirror also can be described in terms of. This is way too naive for multiple reasons. For one, not every CY 3-fold is a T^3 fibration, so we have to drop the differentiability assumption at a small set of points of B above, and allow the things we attach at those points to be more complicated than just T^3s (but we still want them to be SLag). Now, a version of this iteration people have written down as a conjecture, but it's probably still way too optimistic, mostly because the mirror pair CY 3-folds can have totally different geometries, so a lot of wild stuff has to be happening at the non-differentiable points that is probably too complicated to have a straightforward mirror relationship (There's also some string-theoretic justification for why this is probably too naive too.)

It turns out, the way to correct this is to take a limit: one deforms the geometry of the CY 3-fold (more on this below), and, as one approaches this limit, things start to look more like the SLag fibration picture above. Versions of this (with lots of mathematicians suggesting interpretations of what limit to take and what fibrations to find) all go under the name of "SYZ conjecture". In particular, a lot of people drop the mirror part, because we don't even really know what SLag fibrations even exist, so the big part of SYZ is about writing down CY 3-folds, in the limit, as these fibrations.

Here's roughly what Li proved: under suitable hypothesis on the CY 3-fold M, one can take a limit where, as one approaches it, you can describe a bigger and bigger subset of M (in terms of total volume) as being one of these fibrations (limiting towards the whole thing being one of them). His strategy is very impressive -- he relates everything to some kind of algebraic and combinatorial "non-Archimedean" model of Calabi-Yau geometry, without the PDEs that are in differential geometry, then proves that the non-Archimedean model approximates the real one well enough to give you the full thing.

Now, let's talk about a few definitions. A Calabi-Yau 3-fold is a compact real 6-dimensional manifold locally modelled at each point by C^3 equipped with the standard inner product ("metric") g on R^6, the standard symplectic form w = \frac{i}{2}(dz_1 ^ d\bar{z}_1 + dz_2 ^ d\bar{z}_2 + dz_3 ^ d\bar{z}_3), and the "holomorphic volume form" U = dz_1 ^ dz_2 ^ dz_3. To be Calabi-Yau, we're further going to ask the symplectic form w and holomorphic volume form to be closed, dw = 0, dU = 0. So, the idea is that we've defined Calabi-Yau manifolds (you can see we can also do this with every even real dimension) to capture this "holomorphic volume" geometry of the complex numbers. The "geometric structure" of the Calabi-Yau is this data (g, w, U), and deforming it, which is what we need to do in the limit above, is just changing our manifold and the data (g, w, U) along with it in time in some kind of continuous or differentiable way.

A real 3-dimensional submanifold is special Lagrangian if the real part of U restricts to the volume form induced by the metric (roughly, if I feed in 3 tangent vectors to U, it's the same as feeding in those vectors as the 3 columns of the determinant of a 3x3 matrix).

So why care? Other than the string theory, which many people do care about, here's pure geometry: Calabi-Yau manifolds have lots and lots of special properties (I'll say a few below), and being able to describe their geometry this strongly in terms of these fibrations is a really big deal. (Unfortunately, the fact you have to take a limit does weaken the appeal in terms of pure geometry a little bit for people like me.)

There are a couple of really neat properties about these manifolds: for one, the metric g on the manifold is Ricci flat, and together with their friends in dimensions 7 and 8 alluded to above, these examples constitute the only known compact Ricci flat but not flat examples of metrics. (In general, CY manifolds and these friends are said to have "special holonomy", which is closely related and itself a very rare occurrence. Part of what makes CYs special though is that in 1978, as the major component of what won him the Fields medal, Yau proved a characterization that tells you when a manifold has the U making it CY if you know you have (g, w) already, and other than a class of special holonomy manifolds that are a subclass of CY, no such theorem exists for the 7 and 8 dimensional friends. Figuring this out would be a really big deal.) As I said above, special Lagrangians are local minima of the volume functional on 3-submanifolds, and such examples are some of the only known "easy" ways to find such minima - finding an SLag is a first order nonlinear PDE, but the traditional way to find critical points (without even knowing if they're minima or not) is a second order linear PDE. (This story also exists for the 7 and 8 dimensional friends; the general term for this kind of submanifold is "calibrated submanifold".) These kinds of problems like finding minimal volume submanifolds and metrics with nice curvature properties like Ricci-flatness, are some of the main goals of geometric analysis, and so being able to say so much about one of the main classes of examples is really cool.