Why does the periodic signal in time always give a discrete frequency spectrum
Here's an intuitive explanation if the convolution theorem is taken for granted:

Since the time-domain signal is periodic, one can say that it can be built by "copying and pasting" the same block of signal every period: your periodic signal can be expressed as a little block of signal (spanning one period) convolved with a dirac comb.

Thus, its Fourier transform will be the Fourier transform of the little block multiplied by the Fourier transform of a Dirac comb (which is another Dirac comb). Multiplying a continuous signal by a Dirac comb yields a discrete signal.

The same reasoning is also true the other way round (discrete in time implies periodic in frequency).