Honestly, it's not so bad. It's easy to pick any such attempt apart. This is close to my favorite pithy way of explaining it, too, which is to break it down component-wise using the idea of filter banks. It's not a single sentence, but here's what I tend to say:
Any signal—like sounds or electrical signals, or even images—can be thought of as having a certain amount of 'energy' at any choice of frequency. This makes the most sense in music where we might thing of a 3-note chord as having 3 distinct packets of energy at 3 different frequencies.
For any given frequency, we can compute the amount of energy a signal contains at that frequency by comparing the signal with a test signal, a "pure tone" at that frequency. Pure tones are signals that have the unique property of putting all of the energy at exactly one frequency. The Fourier Transform is an equation which packages this idea up, showing us how to represent all of these measurements of energy at all frequencies.
The natural idea of a "pure tone" might be a sine wave. This is what we think of when we think of a musical pure tone and it certainly exists only at a single frequency. But sine waves make for bad comparison signals due to the problem of "phase": two sine waves played together can perfectly support one another and become twice as loud, or they can perfectly interrupt one another and become silence. This happens because sine waves oscillate between positive values and negative values and positive things can cancel out negative things.
When you look at the equation for the Fourier transform you'll see an exponent of a complex number. This is an improved version of a 'pure tone' which avoids the phasing issues of a sine wave. It does this by spinning like a clock hand in two dimensions, remaining always at the same length. This extra dimension lets us preserve enough information so that things never cancel out like with sine waves.