the ABC special did a pretty good job of explaining the basics.
In the case of the Yellowstone Caldera, it is thought to overlie a "mantle hotspot", as does Hawaii, where a particularly strong upwhelling brings heated magma towards the surface. As the magma body rises through the continental crust, it partially melts the crust, adding significant amounts of quartz, and other silica-rich minerals, such as muscovite mica and orthoclase feldspar.
From the above-linked USGS webpage:
"...Scientists infer that rhyolite lava flows as well as the caldera-forming ash-flow tuffs were fed from shallow magma chambers filled by the melting of rocks of the lower continental crust below Yellowstone. The heat needed to facilitate the melting process was supplied by the repeated injections of basalt magma from the mantle into the shallow crust."
As mentioned earlier, these silica-rich minerals make the magma more viscous. As the magma rises, the pressurized fluids (mostly mineral rich water) "wants to boil". Even though more silica-rich eruptions are the norm, from time to time, basalts can also erupt. The basalts are derived from the mantle below and when erupted, it is because the basalt has moved through the crust quickly enough not to melt much of the crust. [Note: The higher-temperature basalts can easily melt the silica-rich minerals,which crystalize at lower temperatures, than do the minerals in the basalts.]
These are large magma bodies, termed "batholiths" that become magma chambers for the overlying eruptions when they reach shallow-enough depths. More than 100 square miles is usually bulged by the rising magma.
When the buoyancy (sp.?) and the gas pressure of the magma overcomes the strength of the overlying rock, as depicted on the program last night, eruptions will break through in several places. These multiple eruptions taking place at the same time will produce gas-driven pyroclastic flows and partially drain the magma chamber. The magma chamber will then collapse inward, opening up a circular to round "ring fracture" system, which will provide more conduits for molten rock to reach the surface, likely triggering even more eruptions. Some of the eruptions may be restricted to the collapsed crater, while others will erupt outward.
Watching the pyroclastic flows from the smaller composite volcanoes gives an idea of the speed of these types of eruptions, but the caldera eruptions may be two to three orders of magnitude larger.