Why is Fast Fourier Transform applied to raw EEG data?

I am trying to understand why Fast Fourier Transform (FFT) is used in the analysis of raw EEG channel data.

My understanding (at the 30,000 ft view) is that FFT decomposes linear differential equations with non-sinusoidal source terms (which are fairly difficult to solve) and breaks them down into component equations (with sinusoidal source terms) that are easy to solve. It then combines each of these component/partial solutions to solve the original equation. In laymen's terms, it's like taking a smoothie, and breaking it down into its recipe of ingredients. Doing so would allow us to study and analyze each ingredient as it relates to the final product (the smoothie).

But how does this relate to raw (non-FFT-decomposed) EEG data? My understanding of a single channel of EEG data is that it is essentially a measurement of voltage over time. What is to be gained by using FFT to break this voltage/time signal down into its constituent frequencies (alpha/beta/gamma/delta/mu waves), etc.? What additional information can each frequency tell us about the raw data?

Update:

Here is my updated understanding of how FFT relates to raw EEG data per @Mark's description below: