How to interpret frequency response of moving average system in discrete time signal processing?
In discrete time, a frequency of zero is just a constant sequence, in complete analogy with continuous time. The main difference between discrete time and continuous time is that in discrete time you have a maximum frequency, whereas in continuous time you (theoretically) don't have one. That maximum frequency in discrete time is achieved for a sequence with alternating values, i.e., the sequence $(-1)^n$ has a maximum frequency of $\omega=\pi$. There can be no larger frequency.
If the sequence was obtained from sampling a continuous-time signal with a sampling frequency $f_s$, then $\omega=\pi$ translates to an actual frequency in Hertz via $f=\omega/(2\pi f_s)$.
Returning to the example of the discrete-time lowpass filter, from the frequency response you see that higher frequencies (up to the maximum frequency $\omega=\pi$) are attenuated. That means that a discrete-time sinusoid's amplitude is changed according to the magnitude of the filter's frequency response at the sinusoid's frequency.
So if the input signal equals
$$x[n]=A\sin(n\omega_0)\tag{1}$$
the amplitude of the corresponding output signal is given by $A|H(e^{j\omega_0})|$. If $\omega_0$ happens to be a multiple of $2\pi/5$ then the input is completely suppressed because these frequencies are the zeros of the filter's frequency response.