the differentiation rule for transforms
Here is the single fact that makes transforms worth the trouble. Differentiation is hard: it is a limit, a local operation, the thing that turns nice equations into differential equations. Under the Fourier transform, that hard operation becomes the easiest thing in the world — multiplication by a number. Differentiating in x corresponds to multiplying the transform by i*xi. That single swap is why a PDE in x turns into an algebra problem in xi.
Precisely: if f-hat(xi) is the Fourier transform of f(x), then the transform of the derivative f'(x) is i*xi times f-hat(xi); the transform of the second derivative f''(x) is (i*xi)^2 f-hat(xi) = -xi^2 f-hat(xi); and in general the n-th derivative picks up a factor (i*xi)^n. You can see why with one integration by parts: integral of f'(x) e^(-i*xi*x) dx, integrated by parts and using that f decays at infinity, throws the derivative onto the exponential, and d/dx of e^(-i*xi*x) is -i*xi e^(-i*xi*x) — out comes the factor i*xi. The decay of f at infinity is what kills the boundary terms and makes the rule clean.
For PDEs the payoff is immediate. Transform u_xx in space and it becomes -xi^2 u-hat. So the Laplacian, the messiest operator in the subject, becomes plain multiplication by -|xi|^2 in frequency. The heat equation's u_t = k u_xx becomes u-hat_t = -k*xi^2 u-hat, an ODE; the wave equation's u_tt = c^2 u_xx becomes u-hat_tt = -c^2 xi^2 u-hat, a harmonic oscillator for each frequency. The number that multiplies u-hat, as a function of xi, is the symbol of the operator, and reading off that number is how you obtain the dispersion relation.
Transform the wave equation u_tt = c^2 u_xx in x: the right side becomes c^2 (i*xi)^2 u-hat = -c^2 xi^2 u-hat, so u-hat_tt = -c^2 xi^2 u-hat. Each frequency xi oscillates like a spring with angular frequency omega = c|xi|, giving u-hat(xi,t) = A cos(c|xi| t) + B sin(c|xi| t). Inverting and combining recovers d'Alembert's left- and right-moving waves.
Each x-derivative costs one factor of i*xi, turning a PDE into an ODE in the frequency.
The rule relies on the function (and enough of its derivatives) decaying at infinity so the boundary terms from integration by parts vanish. For data that does not decay you move to distributions, where the same i*xi rule still holds but needs the wider theory to justify it.