Derivation of Nyquist Frequency and Sampling Theorem
It really boils down to aliasing.  In continuous-time, if you have any two signals $x_1(t) = \sin(2 \pi F_1 t)$ and $x_2(t) = \sin(2 \pi F_2 t)$, then as long as $F_1$ and $F_2$ are distinct, the signals are, too.

But consider sampling at some time interval $T_s$, so that the sampled signals are $x_1(k) = \sin(2 \pi F_1 T_s k)$ and $x_2(k) = \sin(2 \pi F_2 T_s k)$.  If you have a condition where $2 \pi F_1 = 2 \pi F_2 \pm 2\pi n$ for any integer $n$, then the signals are not clearly not distinguishable, by the simple trigonometric identity $\sin \theta = \sin \theta + 2\pi$.  Do the math, and you find that the frequencies cannot be separated by more than the sampling rate to be distinguishable.

It's worse, though, because you also have the trigonometric identity $\sin \theta = -\sin \theta + \pi$.  When you consider that the only thing that allows you to distinguish one signal from another is the frequency, that means that if $F_2$ is a mere $\frac{1}{2 T_s}$ away from $F_1$, then $x_1(t)$ cannot be distinguished from $x_2(t)$.

This phenomenon of two distinct signals appearing as one after sampling is called aliasing.  It's pretty much the basis of the Shannon-Nyquist sampling theorem, and the Nyquist rate.  Because if you're sampling at $\frac{1}{T_s}$ and trying to distinguish signals further apart than $\frac{1}{2T_s}$, you just can't.

Hence, the Nyquist rate.