Mostly it is a concern over the stability of the chloramphenicol, the bacteriostatic agent in the eye drops. It is quite sensitive to temperature degradation, which means it becomes less effective more quickly, and its shelf life consequently becomes reduced should it be stored beyond 8 degrees celsius for extended periods of time. Of secondary concern is that the degradation breakdown products of chloramphenicol are potentially toxic. This is less of an issue, given the relative size of the dose contained within 1 drop, but all things being equal it is better to keep it in a fridge.
Plus which, it feels much more soothing on the eyes, i feel, having a refrigerated drop going in, rather than a room temperature one :)