Why band-limit a signal?
A discrete-time signal represents a continuous-time signal that is band-limited to at most half the sampling frequency. Sampling a continuous-time signal that contains higher frequencies than half the sampling frequency will result in aliasing: Samples of those high frequencies are identical to and cannot be discerned from samples of other frequencies below half the sampling frequency. To avoid contamination of the signal by the aliased frequencies, you would first band-limit (lowpass filter) the continuous-time signal and only then sample it.