I had difficulty seeing the "book solution", but here is, I think, a plausible explanation.

Suppose one of the players is Fred, and let's see if we can find the probability that he gets all 4 aces. For convenience of analysis, let's say the first 13 cards in the deck are dealt to Fred. If we simply distinguish between aces and non-aces, and we consider the aces indistinguishable, then there are

possible locations of the aces in the deck, all of which we assume are equally likely. Of these,

have all the aces among the first 13 cards. So Fred's probability of getting all the aces is

.

Multiply by 4 to obtain the probability that some player, not necessarily Fred, gets all the aces.