Basically, it's a consequence of negative heat capacity.
Gravitationally bound systems can (often do) behave such that adding energy results in reducing temperature, and vice versa. You can understand this intuitively if you consider a simple two-body system: adding energy to the system causes the orbits to expand, and bodies on larger orbits move at lower speeds; thus, even if the orbits are highly eccentric, the average speed of the two bodies goes down when energy is added, and this generalizes to systems of many particles, where the average velocity determines the temperature of the system.
If a system with negative heat capacity comes into contact with a large thermal reservoir at higher temperature, it will absorb heat from that reservoir... and get colder. And thus continue absorbing heat until the reservoir is exhausted- it cannot come to equilibrium.
If the higher-temperature reservoir is another gravitationally bound system (in particular, if it is a subset of the same gravitationally bound system), then you have the conditions for a gravothermal catastrophe. This is because a high-temperature, negative-heat capacity system in contact with a cold sink will give up heat until it is entirely exhausted, getting hotter, rather than colder, as it does so. So, when you have a high-temperature gravitational system and a low-temperature gravitational system in contact, you end up with all of the energy being transferred out of the hot section, which contracts, and into the cold section, which expands, and probably becomes unbound.
This tends to happen in old globular clusters and bright elliptical galaxies, where the cores can get hot, start transferring kinetic energy to the outer regions, and collapse.