I don't understand the remark 3 following the statement of the theorem 7.4 p.185 of "Functional Analysis, Sobolev Spaces and Partial Differential Equations" (2011), by H. Brezis.
Here the theorem takes the following form: $\exists !\ u$ function from $\mathbb{R}_+$ with value in a Hilbert space $\mathcal{H}$, with certain regularity, statisfying $$ \left\lbrace \begin{aligned} & \frac{d}{dt} u + Au =0\\ & u(0)=u_0 \end{aligned}\right.$$ where $A$ is an operator such that $\forall\ f\in \mathcal{H},\ \exists\ u\in D(A)$ s.t. $u+Au =f$.
The remark then says "The main interest of Theorem 7.4 lies in the fact that we reduce the study of an “evolution problem” to the study of the “stationary equation” "u + Au = f" (assuming we already know that A is monotone, which is easy to check in practice)"
By evolution, does he just mean an equation of the form $u - \frac{d}{dt}u =f$?
"Evolution" refers to something that changes with time. So the differential equation where the derivative is with respect to time is an "evolution problem". If the differential equation $u'+Au=0$ describes some system, the solution $u(t)$ describes the evolution of the system.
The Hille-Yosida theorem guarantees that the above differential equation has a solution if we know that $A$ is maximal monotone. And checking that $A$ is maximal monotone has nothing to do with differential equations; it is about checking that $u+Au=f$ has a solution for every $f$, no derivatives involved.