Fourier Transform of Impulse Train
Intuition can sometimes be misleading.  But here are some ideas that might help one move towards creating a mental picture.

An infinitely long pure sinewave in the time domain (consisting of just one frequency FT or DFT basis function) will be a single impulse in the frequency domain.

Distort the sinewave a little, but leave the waveform perfectly periodic, and the impulse will be followed by an evenly spaced harmonic series.  Usually the narrower and sharper the distortion (but keeping the waveform still perfectly periodic), the longer the harmonic series.  What might be considered a limiting case of maximum distortion, the narrowest waveform with the sharpest edge will have the longest harmonic series.  Or an infinitely long sine wave in the time domain maximally distorted into just an infinitely periodic impulse train, will produce a impulse followed by an infinitely long harmonic series, which looks a lot like another periodic impulse train.

Make it an even (cosine) function, and all the impulses in the FT will be real and thus symmetric around 0.  Add a DC offset to the distorted sine wave to complete the pulse train at 0.