Real life is making this week very busy, so it will be hard for me to write much in a timely way about this, and the brief popular version by the Nobel Foundation is pretty good if you're looking for an accessible intro to the work that led to this. Their more technical background document (clearly written in LaTeX) is also nice if you want greater mathematical sophistication.
Here is the super short version. Thouless, Kosterlitz, and Haldane had major roles to play in showing the importance of topology in understanding some key model problems in condensed matter physics.
Kosterlitz and Thouless (and independently Berezinskii) were looking at the problem of phase transitions in two dimensions of a certain type. As an example, imagine a huge 2d array of compass needles, each free to rotate in the plane, but interacting with their neighbors, so that neighbors tend to want to point the same direction. In the low temperature limit, the whole array will be ordered (pointing all the same way). In the very high temperature limit, when thermal energy is big compared to the interaction between needles, the whole array will be disordered, with needles at any moment randomly oriented. The question is, as temperature is increased, how does the system get from ordered to disordered? Is it just a gradual thing, or does it happen suddenly in a particular way? It turns out that the right way to think about this problem is in terms of vorticity, a concept that comes up in fluid mechanics as well (see this wiki page with mesmerizing animations). It's energetically expensive to flip individual needles - better to rotate needles gradually relative to their neighbors. The symmetry of the system says that you can't spontaneously create a pattern to the needles that has some net swirliness ("winding number", if you like). However, it's relatively energetically cheap to create pairs of vortices with opposite handedness (vortex/antivortex pairs). Kosterlitz, Thouless, and Berezinskii showed that these V/AV pairs "unbind" collectively at some finite temperature in a characteristic way, with testable consequences. This leads to a particular kind of phase transition in a bunch of different 2d systems that, deep down, are mathematically similar. 2d xy magnetism and superconductivity in 2d are examples. This generality is very cool - the microscopic details of the systems may be different, but the underlying math is the same, and leads to testable quantitative predictions.
Thouless also realized that topological ideas are critically important in 2d electronic systems in large magnetic fields, and this work led to understanding of the quantum Hall effect. Here is a nice Physics Today article on this topic. (Added bonus: Thouless also did groundbreaking work in the theory of localization, what happens to electrons in disordered systems and how it depends on the disorder and the temperature.)
Haldane, another brilliant person who is still very active, made a big impact on the topology front studying another "model" system, so-called spin chains - 1d arrangements of quantum mechanical spins that interact with each other. This isn't just a toy model - there are real materials with magnetic properties that are well described by spin chain models. Again, the questions were, can we understand the lowest energy states of such a system, and how those ordered states go away as temperature is increased. He found that it really mattered in a very fundamental way whether the spins were integer or half-integer, and that the end points of the chains reveal important topological information about the system. Haldane has long contributed important insights in quantum Hall physics as well, and in all kinds of weird states of matter that result in systems where topology is critically important. (Another added bonus: Haldane also did very impactful work on the Kondo problem, how a single local spin interacts with conduction electrons.)
Given how important topological ideas are to physics these days, it is not surprising that these three have been recognized. In a sense, this work is a big part of the foundation on which the topological insulators and other such systems are built.
Original post: The announcement this morning of the Nobel in Medicine took me by surprise - I guess I'd assumed the announcements were next week. I don't have much to say this year; like many people in my field I assume that the prize will go to the LIGO gravitational wave discovery, most likely to Rainer Weiss, Kip Thorne, and Ronald Drever (though Drever is reportedly gravely ill). I guess we'll find out tomorrow morning!