I expect that strength increases off-cycle are due to biological effects often being delayed relative to receptor activation. For example, suppose that under the influence of high androgen levels, a satellite cell fuses with a mature muscle cell, thus increasing its number of nuclei. This will result in ongoing higher capacity for protein synthesis, but the muscle wouldn't have the full benefit or anything like it on the first day of this occurring. It might take quite a while.
It also may take the body time to learn to use the increased capacity of the muscle for strength. Even though it might be the case that the same nerve impulses will now produce a more forceful contraction, initially one might not have the same nerve impulses activating the muscle, as protective mechanisms may cause the higher weight to seem heavier than can be lifted.
My experience and that of anyone I've asked what they thought on it is that as one gets stronger, it is not that weights feel say half as heavy on getting twice as strong. No, a much heavier weight still feels much heavier: it's just that now you can lift it despite it feeling much heavier. But I would suppose that if somehow, magically, that example doubling of strength happened overnight, for quite some time your body would not let you do it as it felt impossibly heavy.
The same thing might be the case, though of course at much smaller percentages, with the increased capacity for strength generated in the "on" weeks. Perhaps not everything that is in fact there at that point is yet utilizable, but in the following weeks the body learns to employ it.