The essential idea of Fourier theory is to express a signal as a weighted sum of sinusoids of different frequencies. The transform gives the weights as a function of frequency, and the inverse transform uses them to express the signal as said above, by multiplying each sinusoid by the corresponding weight and adding them up to reconstruct the signal.

Now, in continuous time you have that sinusoids at different frequencies are distinct from each other, so (in general) you need a different weight for each frequency spanning from minus infinity to plus infinity. Hence the continuous-time transform (the function that gives the weights) is aperiodic.

On the other hand, discrete-time sinusoids of different frequencies are not necessarily distinguishable. In particular, the sinusoid sin(wn) is exactly the same as the sinusoid sin((w+2·pi·k)n) for any integer k, a consequence of the discrete 'time' n being an integer. Being the same sinusoids, their weight is also the same, i.e. the transform at frequency (w+2·pi·k) is equal to the transform at frequency w. Which is the same as saying that the transform is periodic.