Fourier transform diagonalizes time-invariant convolution operators
TL;DR: Fourier is the logarithm of the convolution.
Doing a convolution (linear time-invariant operator) is a quite involved processing in the original (primal) domain: it involves intricate sums and products of an operator $h$ and signals $s$ at different indices:
$$ \sum_k h[k]\times x[n-k]\,.$$
With a Fourier transform, you transform the whole information for the operator $h$ and data $x$ in the primal domain onto a novel dual (frequency) domain, in $\hat{h}$ and $\hat{x}$. Then the result of convolution becomes:
$$  \hat{h}[m] \times \hat{x}[m]$$
which is a simple index-by-index product. Hence the term &quot;diagonalization&quot;: from products of two indices, $[k]$ and $[n-k]$, Fourier turns them into a product of a single index $[m]$ times  $[m]$.
To make the above a bit simpler, Fourier is to convolution what logarithms are to multiplication. Let us forget the carry for a moment. Doing a multiplication is complicated, especially for large numbers: one has to multiply thousands to tens and add them to products of hundreds.  Logarithms turn multiplication into adds: you (almost) have to add figures in the same position (lest the carry), instead of combining figures in different powers.
For ages, people have used logarithm tables or logarithmic sliding rulers as analog computers. Logarithmic tables and rulers have been very important in days before computers, to allow computations in engineering, astronomy, etc.
Fourier principles, tools and fast librairies are the nowadays analogs of logarithms, that helped shaped our digital world. The FFT algorithm is often considered one of the most important algorithms. Indeed, there exists FFT based multiplication of large numbers.