OFDM - cyclic prefix ?

There are two different definitions of spectrum. Both are relevant in their
own respect, but not the same.

As far as the FFT is concerned: It "sees" one cycle of a signal that might
be periodic from minus to plus infinity. There is ideally no energy leakage
from one subcarrier to the other, including for example adjacent but unused
frequency regions inside the same FFT.

The other definition becomes relevant when I connect a radio frequency
spectrum analyzer (or its baseband equivalent). 
Other than the FFT, it knows nothing about the cyclic assumption, and it
is not synchronized with the symbol structure.
Suddenly the spectrum looks quite messy, energy seems to leak everywhere.

That is caused by the DISCONTINUITIES between symbols: At the symbol
boundary, the signal "jumps" from one instantaneous amplitude to another.
This can be expressed as multiplication of each symbol with a rectangular
window => sinc spectrum. It is completely transparent to the FFT, but
hurts other users that aren't synchronized and can't take advantage of the
single-cycle processing.
In an implementation, an "ideal" symbol-to-symbol discontinuity from a
rectangular window jams too much energy into adjacent bands, and we need
to tame it somehow (for example filtering)

Hope that clarifies somewhat.

-Markus