One way to think about it is the requirement of what a filter does, and what is the relation between the time domain and frequency domain plots of the signal or the filter.
This also requires to know the fact that multiplication in frequency domain is identical to convolution in time domain.
If you want to make a filter that passes one specific frequency, say 1 kHz, this is easy to think in frequency domain as there is only one frequency and that's at 1 kHz at amplitude of 1. If you multiply this in frequency domain with your signal, it only leaves the 1 kHz component of your signal in the result. Now it is important to realize that the filter is simply a sine wave at 1kHz in the time domain, so convolving the original signal with sine wave of 1 kHz will only leave the 1 kHz frequency and will remove other frequencies.
So, as we know the shape of a brick wall low pass filter in frequency domain is a step function, we know that all the infinitely many frequency components that are needed to pass have a coefficient of 1 up to the cut-off frequency and the other components have a coefficient of 0 so they don't exist.
With this in mind, we see that the time domain signal is a sum of basically infinite amount of sine waves at all frequencies from 0 up to the cutoff frequency.
So if you sum up all the infinite amount of individual sine waves of different frequencies between 0 Hz and cut-off frequency, you end up with the time-domain waveform of the filter. And it happens to be the sinc waveform.
This sum of infinite amount of sine waves up to the cut-off frequency can be proved from Euler's work to be of the sinc, and also seen through fourier transform that a rect shape in frequency domain is sinc shape in time domain, and also that the rect shape in time domain is sinc shape in frequency domain.
So, in short, sinc shape is just the shape of a time domain waveform that contains infinite sum of sine waves at all frequencies from 0 to the cut-off frequency, which convoluted with the time domain signal will keep all frequencies between 0 Hz and cut-off and remove all frequencies above cut-off as it did not have those frequencies.