The explanation for the appearance of collapse lies in the phenomenon of decoherence, which basically says that subsystems tend to quickly evolve into something resembling an eigenstate. This evolution must necessarily occur on an incredibly short timescale. It might be possible to design an experiment that would test the assumption that collapse is instantaneous.
I think the best definition of "collapse" is that it is the moment in time in which a particular system can no longer be described (to good approximation) as the direct product of two subsystems (see http://en.wikipedia.org/wiki/Separable_state). The concept of a "good approximation" is of course subjective, but it can always be objectively metricized (totally made that word up) by using some kind of error term.
Epigrammatically, collapse is not so much a physical process as it is a characterization of the capability to represent a quantum state in a specific mathematical form.
(Of course, this doesn't preclude you from categorizing physical processes as "collapse events"; it just means that collapse isn't a fundamental phenomenon so much as it is an emergent one. Kind of like quasiparticles.)
So what about the randomness? I think it's better to refer to it as unpredictability. The difference is subtle but crucial. True randomness (assuming it exists) is the result of absolute indeterminism. On the other hand, if eigenstate selection is merely "unpredictable", then that implies collapse is in fact a deterministic process (specifically e^(-iHt) applied to Ψ over some time interval that we've decided to call a "measurement"); however, we're unable to extract enough information from the environment to make exact predictions because we ourselves constitute the required missing information. In other words, the information necessary for absolute predictive capability is trapped in the subsystem constituting the measuring environment, and it becomes lost when that subsystem becomes entangled with the subsystem being measured. And there's not really any way to prevent that from occurring, because entanglement must occur in order to learn anything about a system.
This even applies classically. The only difference is that classical entanglement occurs between localized physical boundaries instead of between subspace boundaries in an abstract Hilbert space.
To somewhat reify this, assume (for the sake of argument) that a classical description of physics is enough to describe a human. Then perform a large MD simulation of all the atoms inside a physics lab, including those of a physicist. The evolution of this simulated system is provably deterministic. Yet the physicist appears to have free will, and it appears like he is deciding which measurements to perform on his environment. But he's just an arbitrary collection of atoms that we've labeled "human", and he obeys the same time-transformation rules that the unlabeled atoms in the system obey. Mathematically, it's simply impossible for him to predict everything that occurs within the virtual system -- not because of indeterminism -- but because he isn't so much "choosing" what to measure as he is "appearing to choose". There's a limit to the amount of information any system can obtain about itself (well, maybe there's some fractals that are exceptions, but generally speaking, it holds true.)
That said, experiment is always the ultimate arbitrator of truth, and I wonder if there might yet be some clever way to tell whether our universe is simply unpredictable instead of random, despite the possibility that both potential mechanisms might impose the same limits on predictive capability (in fact, Colbeck and Renner recently proved that QM is already maximally predictive, independent of whatever underlying mechanism governs eigenstate selection -- see http://www.nature.com/ncomms/journal/v2/n8/abs/ncomms1416.ht...)