A phase transition is when the behavior of a material changes in an abrupt, qualitative way, say between being liquid and being solid, or between being a permanent magnet and being demagnetized. The point of transition is called the critical point, and phase transitions are also called critical phenomena. Abrupt, qualitative changes in dynamics, like going from a stable equilibrium to oscillations, are called bifurcations. Much of what follows for phase transitions applies, mutatis mutandis, to bifurcations.

Baring what can best be described as improbable mathematical conspiracies, the behavior of a material near a critical point is determined by only a small number of qualitative parameters --- the number of dimensions the system has, its symmetry properties, and so on. That is, if two materials have the same values of those parameters, then their behaviors become more and more similar, the closer they approach their critical points. Those materials are said to fall into the same universality class. It further turns out that the number of universality classes is comparatively small, certainly much smaller than the number of phase transitions.

This is where toy models come in. Suppose you want to study the transition from being a permanent magnet to being demagnetized in, say, iron. You could try to work out all the physical interactions in iron, model them, and solve the model near the critical point. This would be a nightmare. Alternately, you could take what you know about the symmetries and build a toy model which also has them. If you do it right, the toy model will be in the same universality class as the real material. You then solve the toy model near the phase transition, and you get accurate predictions about the real phase transition --- about all the real transitions in that universality class. All the details that make the difference between the interactions in the real materials, and the stylized ones in the toy model, don't make any difference, near the phase transition.

It's easy to see why statistical physicists are very fond of this method. It is also easy to see why they sometimes forget that this only works sufficiently close to a critical point --- that away from criticality, details matter intensely.

Needless to say, most of the things Wolfram is talking about are not close to phase transitions, so universality-class arguments won't save his toy models.

For a relatively gentle introduction to phase transitions and universality, see Julia Yeoman's The Statistical Mechanics of Phase Transitions. I'm sure there are popular treatments, but I don't know of them.