For example, the Banach fixed-point theorem is applied in the proof of the Picard–Lindelöf theorem about the uniqueness of solutions of ordinary differential equations and the Lefschetz fixed-point theorem (or a modification of it) is used in the proof or in the context of the Weil conjectures. There are so many more examples.
What is so special about the equation $f(x)=x$?
On
Suppose we have a dynamical system obtained by iterating a continuous mapping: $x(n+1) = f(x(n))$. We are typically interested in what happens in the limit as $n \to \infty$. Perhaps the simplest behaviour we might hope for is that $x(n)$ goes to some limit $L$ as $n \to \infty$. Taking $n \to \infty$ in the equation $x(n+1) = f(x(n))$ and using continuity of $f$ gives us $L = f(L)$, so $L$ should be a fixed point of $f$.
Of course an equation $g(x) = 0$ can be transformed to a fixed-point equation $f(x)=x$, e.g. by taking $f(x) = g(x) + x$. What can be nontrivial, though, is the use of the dynamical system $x(n+1) = f(x(n))$ to iteratively solve the original equation.
This doesn't explain every appearance of fixed points, but it is an important motivation for some of them.
One thing that immediately comes to mind: it is by far the simplest equation one can write that's 'universal' (and not tautological). All one needs is some domain $D$ with an equality relation on it and a function $D\mapsto D$. $D$ can be discrete; it can be $\mathbb{R}$; it can be infinite-dimensional; it can have extra structure or no structure at all. We don't need to be able to combine elements of $D$ in any meaningful ways, we don't need to compare in any way other than equality.