 When we sample a signal, turning it into a discrete time signal, we are effectively throwing away some of the information about that signal. We only know what it is doing at the point where we sampled it. Between those points, anything could be going on. The DFT tests our signal with cosine and sine waves at many different frequencies to find out which frequencies are present in the signal. However, once the test frequency is high enough so that it has time to complete more than half a cycle between two sampled points, as far as the DFT is concerned, because it doesn't know what is happening between two sampled points, the higher test frequency looks exactly the same as a similar much lower frequency and it thinks the higher frequency is also present in the signal. This is why the frequency domain representation of a discrete time domain signal is periodic around odd multiples of half the sampling rate.