my dynamical system/ ODE textbook says:
"if we consider the solutions of the autonomous system $x'=f(x)$, $x(0)=x_0$, for $f\in C^k(M,\mathbb{R}^n)$ and open $M\in\mathbb{R}^n$, we can regard such a system as a vector field on $R^n$.
question: how can we regard this as a vector field on $R^n$?
The map $f$ can be seen as a vector field on $M$, since for each $x\in M$ you have a vector $f(x)$ at that point.
Strictly speaking this is unrelated to the solutions of the equation (in this respect the sentence in the book is quite misleading), although to solve $x'=f(x)$ can be as finding curves with the property that at each point are tangent to the vector field.