Those equations are especially easy when applied to trigonometric functions like sine and cosine (because their derivates are easy to express in terms of each other).
He didn't really have a way to solve the equations for other functions directly, but he tried to approximate them by sums of trigonometric functions. He noticed that he could actually do that for all functions he wanted to analyze, and wrote about that in his book.
Later, other mathematicians qualified the circumstances that make such a transformation possible, formally proved its properties and named it "Fourier transform" in his honor.
Because of its applicability to problems involving differential equations, the Fourier transform was also used in other domains of physics e.g. to analyze vibrations like sound.
Because of those many uses, people needed a way to compute it quickly, so they optimized the algorithm to create the Fast Fourier Transform.
So those "tools that are so important in digital signal processing today" were not invented by Fourier singlehandedly, but are associated with his name due to him kicking off the initial development.
Actually, Gauss used a variant of the FFT in his astronomical work even before Fourier published his own results, but because Gauss didn't publish, he is not associated with the discovery.
I assume that except for the adaptation to modern notation, the general idea of the argument has been preserved.
Note that sinusoids are only used because they make the calculations convenient, and not because of the day/night cycle or actual sinusoidal heat distributions empirically observed in nature.
George boole lived in 18th century. He came up with boolean algebra which is used extensively in digital circuits.
Euler, Newton, and Gauss all lived from 16th - 18th century and they are responsible for a large chunk of the math we have today which a reason we have computers.
We stand on the shoulder of giants.
