As with all questions about why animals do any specific thing, the ultimate answer is simply "those that didn't died out". So the question of why do birds that bob their heads have an evolutionary advantage over those of their species that didn't does not necessarily need or have a single absolute answer.

In this particular case, three major proximate reasons (any of which might have more or less impact in any particular case) are:

• tracking: the mechanical movement is actually lack of movement, because keeping their eyes nearer a fixed level is less energy-intensive than using their brains to compensate for the movement of their heads while walking

• balance: to lower their center of gravity to make toppling over when they lift one leg to step forward less likely

• leverage: by moving their heads in an opposite direction to their bodies, they use less energy in locomotion

Thanks for your time. Hope it helps.