A non-rigorous but intuitive explanation would be to note that for zero-mean (wide-sense) stationary processes, the autocorrelation at lag $\tau$ is the correlation between two samples of the process at a temporal distance $\tau$. It seems natural that with certain regularity assumptions (intuitive: "sufficient randomness"), that correlation should decrease (on average) with increasing lag, and should finally converge to zero for $|\tau|\rightarrow\infty$.
An example of a process for which this is not the case was given in Fat32's answer, but such a process lacks what I hand-wavingly referred to as "sufficient randomness", because it can be parameterized by a finite number of random variables. Such a process is sometimes called singular. Singular processes have Dirac delta impulses (at $\omega\neq 0$) in their power spectrum. Note that for the given example, the limit of the auto-correlation function does not exist.
The power spectrum of a non-singular (regular) process can have only one Dirac delta impulse, and that must be at $\omega=0$, reflecting a non-zero mean. Since the auto-correlation is the inverse Fourier transform of the power spectrum, that delta impulse causes a constant term in the auto-correlation, and, consequently, for a non-zero mean WSS process, the auto-correlation function cannot converge to zero, but it converges to the square of the mean:
However, if $\mu_X=0$, you generally have (for regular processes)
As a side note, since the power spectrum is the Fourier transform of the auto-correlation, the power spectrum can only exist as a conventional function (without Dirac delta impulses) if $(2)$ is satisfied, because otherwise the Fourier integral does not converge in the conventional sense.