The question that Azzie Rogers linked to: How can we determine the size and composition of Earth's inner core?
Does answer the theory part of the question, but I will extrapolate a little more to answer the question.
The question becomes, what kind of waves travel through what? Both shear and compressional seismic waves can travel through solids, but as it turns out, you cannot shear a liquid. As you look at teleseismic raypaths (waves traveling far from the source) you can see that the wave must travel deep into the earth and then back up again. The wave forms, and directions of these wavepaths will be altered by the different compositions of Earth's layers as it travels through: a wave just traveling through the mantle will have both its shear and compressional component. A wave traveling through the mantle and outer core will lose its original shear component, and either develop a new one as it leaves the core (but distinct from the wave that travels through the mantle only) or not have one at all. And finally, a wave that travels through the mantle, outer core and inner core will have distinct wave patters as well. Its by this comparison of waveforms that we realize that the earth not only has different compositional layers, but phase boundaries as well. I am not sure seismology alone would lead to a solid inner core, but there is abundant evidence supporting that fact. When you combine Seismology, rare bits of geochemistry, the calculation of Earth's gravity , and perhaps the most important part, the magnetic dynamo generating our magnetic field, we see a fairly clear ( albeit incomplete) story.
