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First of all, this conveys an inability to have truly grasped the fundamental essence of quantum physics, and beyond that, goes so far as to naively -- or even arrogantly -- generalize human-scale causality to the quantum scale.
What Peikoff is basically saying is that he intuitively knows what is really going on, at the quantum level, and that it's just a matter of time before physicists confirm his beliefs. This, despite no evidence which supports anything but apparent actual randomness of subatomic happenings at any given point in time.
Unfortunately, my copy of OPAR is currently everywhere in my apartment and car simultaneously (no, sorry, I'm just not sure where it is right now), so I can't check the context of the quote, and if Peikoff elaborates on this point any further I can't respond to anything else he says on the matter. But given just this specific quote, I think you're misinterpreting him. He is not saying that he knows magically that quantum entities will exhibit causality. All he is saying here is that the fact that we are unable because of the nature of the tools we currently use for measurement to predict quantum events is not sufficient evidence to reject causality in the quantum world. Quantum events still may not be causal, but before anyone can make such a claim they have to furnish better evidence than our inability to presently predict them.
Now, he may be saying this as part of an a priori claim that quantum events are causal. If he is, he is not entirely unjustified in this either. Everything we have encountered thus far in the universe has exhibited causality; doubting that quantum events do as well without some very convincing evidence is about as silly as claiming that, even though every rock we've dropped thus far has fallen downward at 9.8 m/s^2, we have to consider that the next one might go up with some constant velocity. The burden of proof for a noncausal quantum world is on the advocates of such a world, and to the extent of my knowledge, they have not furnished it.
(Incidentally, simply the fact that scientists have made as much progress as they have in the field of quantum computing seems to indicate that the quantum world is at least sufficiently controllable and predictable for human purposes.)
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