>>rambol+mj
I had a professor describe a FT as a "dot product of a function against the real signal space". Thus a FT is valued higher at frequencies where the input signal is more "similar" or "in line" with that frequency. Conversely, the FT is zero where there are none of those frequencies in the input signal.
If this helps, then it can also help with understanding other projections such as the Laplace transform (a dot project against the complex signal space).
While this analogy has helped me, I still have no clue why real valued signals result in an even FT.
edit: grammar