The better question perhaps, is why is $e^z$ periodic?
Well this has somewhat of a geometric interpretation, and for simplicity lets stick with a complex number of the form $i \theta$. Then:
$e^{i \theta} = \left( \cos \theta + i \sin \theta \right)$
and you can think of this as identifying the point $(1,\theta)$ on the unit circle in the complex plane (this is polar coordinates). But then of course if you add $2 \pi$ to this angle, you just get back to the same point on the circle. Thus the complex exponential function cannot be injective.
Thus if you want the logarithm of $w$ to be a complex number $z$ such that $e^z=w$, then this will also carry a periodicity.