The graphic doesn't really help show how the analysis itself happens, it just presents the result, which is a series of waves that add up to f. The actual process of obtaining a frequency coefficient from a time-domain function is easy to describe: multiply the function by a (co)sine wave with a particular frequency and sum together the result. But it's not very intuitive why that works until you consider that one period of a sine wave sums to zero. By multiplying the sine with the function, you perturb the shape of the sine with just the amount of energy that the function contains at that given frequency. So that instead of summing to zero, the sum measures "how much" of that particular wave is present. That's Fourier analysis.

Fourier synthesis is more easily visualized (I think anyway). Simply multiply a sine wave at each frequency by the corresponding coefficient derived above, and sum those weighted waveforms together elementwise to recover your function.

Thank you. This really helped.

Based on that explanation, wouldn't the formula be more like: http://www.texpaste.com/n/nphm1fgp ? I suppose there is some further magic which allows for phase differences or something.

Transforms that use only sines or cosines (like the DCT) provide a complete basis by increasing in frequency by only a half cycle (pi) rather than by integer cycles (2pi). Essentially trading half a transform of sines and half a transform of cosines for one transform of half-cosines (or sines).