I am twenty years removed from it, but I used to be well-informed on this kind of thing; I have a PhD in mineral physics.
Let's see how well I can explain this (haven't read the article, yet, sorry! Waiting for a plane...)
So you're no doubt familiar with the physics of a vibrating string; it resonates at wavelengths (length of string, 2 * length of string, 3 * length of string... n as n->inf). So you can express any vibrational state of the string as sum(intensity * wavelength); so you can represent the state of the string as a vector of intensities on a basis of allowable vibrations of the string.
Let's call these _vibrational modes_. Let's assume occupancy of these modes is quantized. It's (sort of...) the same as energy levels of atomic orbitals in electronic structure, if you remember that from chemistry classes; the way it's not the same is important (bosons vs fermions) but not at this level of handwaving :-)
So this is how solids store energy, and we call this energy "heat".
A reasonable approximation for a crystalline structure is balls – point masses – connected by springs, where the springs are covalent bonds, plus electrostatic effects between point charges. Intuitively, you can follow that the same kind of _vibrational spectrum_ will arise from this arrangement (in the same kind of way; the solutions of the differential equation of this system of forces under periodic boundary conditions). So materials have resonant frequencies in the same way guitar strings do.
Therefore, this vibrational spectrum defines the thermal behavior of a material; heat capacity, thermal conductance, etc etc etc etc. Each of these vibrational modes is also tied to a collective motion of the particles in the material, which (if sufficiently violent) will tear the structure apart – there's the solid-to-liquid phase transition – or, more subtly, if lost will lower the symmetry of the crystal structure, which gives rise to solid-to-solid phase transitions (an example would be alpha to beta quartz, which will crack your crockery if you leave it in the oven on a cleaning cycle; https://en.wikipedia.org/wiki/Quartz_inversion).
There's a lot of depth here, as you can imagine!