Why are the values of angular frequency of discrete-time sinusoidal signals in between -pi and pi?
Because the sequence
$$ x[n] = A \cos( \omega n) $$
will produce exactly the same samples for all discrete-time frequencies such as $\omega = \pi/3$ or $\omega = \pi/3 + 2\pi$ or $\omega = \pi/3 + 6\pi$ or $\omega = \pi/3 -8 \pi$... In other words every frequency of the form $w = w_0 + 2\pi k $ for all integers $k$ will yield the same set of samples; they are indistinguishable, and we chose to use the range $[0,2\pi]$ or $[-\pi,\pi]$, when referring to the discrete-time sinusoidal frequencies.
Note that the periodicity of the discrete-time frequency $\omega$ relies on the fact that the index $n$ is an integer. Otrherwise, like in the continuous-time case, the frequency $\omega$ would range from $-\infty$ to $\infty$.