The idea that the rear ranks protected the formation from missiles seems to originate with Polybios' description of the Macedonian phalanx:

It will not be difficult to imagine what the appearance and strength of the whole phalanx is likely to be, when, with lowered sarisai, it advances to the charge sixteen deep. Of these sixteen ranks, all above the fifth are unable to reach with their sarisai far enough to take actual part in the fighting. They, therefore, do not lower them, but hold them with the points inclined upwards over the shoulders of the ranks in front of them, to shield the heads of the whole phalanx; for the sarisai are so closely serried that they repel missiles which have carried over the front ranks and might fall upon the heads of those in the rear. These rear ranks, however, during an advance, press forward those in front by the weight of their bodies; and thus make the charge very forcible, and at the same time render it impossible for the front ranks to face about.

-- Polybios, Histories 18.30.1-4

The interesting thing is that other descriptions of this formation do not include this detail. The tactical author Asklepiodotos, writing around the same time as Polybios or slightly later, describes all the same advantages of the pike phalanx - the five ranks of projecting sarisai, the "push of pike" exerted by the rear ranks, and the obstacle presented by the depth of the formation. He lists these advantages in much the same order and language as Polybios. But he says nothing about warding off missiles.

Since the argument seems to be unique to Polybios, it is also relevant to note that he presents it purely as a theory. He is not referring to a specific battle in this passage. He's not saying "this is what I heard from eye-witnesses who saw their arrows glance off the forest of pikes". He offers no example to corroborate the claim. The passage simply lists the general reasons why the pike phalanx was both terrifying and highly effective in pitched battle.

When we consider the similarity between his description and that of Asklepiodotos (using him as an example of the tactical authors in general), it becomes pretty tempting to suggest that the entire notion of sarisai warding off missiles is Polybios' invention. He seems to have based his description of the pike phalanx around the tradition of contemporary tactical manuals, grafting his own theory onto it. Perhaps he wasn't satisfied with their explanation of what the rear ranks were doing (i.e. blocking the retreat of the front ranks). Perhaps he wanted to offer an explanation as to why the rear ranks even carried sarisai if they could not reach the enemy with them. Either way, this unsubstantiated theory has been so widely repeated that it is now part of the received wisdom about how the Macedonian phalanx worked.

As to what the pike phalanx's actual advantages were, I believe the Polybios passage otherwise sums them up nicely. The principle of pitched battle in Antiquity was to drive the enemy off the field; the main way to do this was to break his heavy infantry core or force it to retreat. This was what the pike phalanx was designed to do. In perfect conditions (a flat battlefield with no obstacles and no other troop types present), a Macedonian phalanx was invincible, since its sarisai could reach any type of enemy heavy infantry without allowing those troops to reach its own, and the density of its formation prevented the enemy from breaking it apart and destroying it in detail. It was impossible for any other form of heavy infantry to withstand, let alone drive back, a Macedonian phalanx. Infantry formations facing a pike phalanx had no choice but to retreat or die, and either option would mean victory for the pikes.