At a first sight it looked like a split flap but as @yshavit found out, it is actually a zap flap. A zap flap is a particular kind of split flap that via a leverage moves not only downward but also backward, increasing the chord as well. Apparently the name derives from his prolific inventor, Edward Frank Zaparka, who patented it in the 1933.
Split (or zap) flap together with plain flap have been historically the first iteration of trailing-edge high-lift devices. They have been superseded by more efficient albeit mechanically more complicated flaps, like the three-slotted flaps seen on virtually all modern jetliners. This picture taken from Wikipedia shows the several types of existing flaps.

Why a zap flap instead of a simpler split flap? The following plots taken from this NACA TN show that moving the flap backward as well, slightly increases the $C_{l_{max}}$ practically without any increase in the $C_d$:

On the left side there's a split flap while on the right side there's a zap flap. Due to the backward extension of the flap, the chord i.e.wing surface becomes bigger and bigger surface implies higher lift.
From the plots it can also be seen what is in general the effect of a split (zap) flap: both the plots of $C_l$ and $C_d$ are basically shifted more and more upward the bigger the flap deflection is. With modern slotted flaps the upward shift of $C_d$ is more limited.
But why exactly a zap flap instead of a plain or nother kind of flap? According to the relevant wiki entry, the F-101 suffered from "severe pitch-up at high angle of attack". High angle of attack are seen also during landing phase so a kind of flap which didn't changed too much the wing's pitching moment would have been welcomed: split flaps have just this characteristic. So it might be (just speculating here) that the choice of a split (zap) flap was dictated by this requirement.