In the third edition of Peter Petersen's book Riemannian Geometry, there is a following statement in page 46:
The case where $ X|_p = 0 $ is of special interest when computing Lie derivatives. We note that $ F^t(p)=p $ for all $ t $.
Here $ F^t(p) $ is the local flow such that $ F^0(p) = p $ and $ \frac{\mathrm{d}}{\mathrm{d}t}F^t(p)|_{t=0}=X_p=0 $.
But how could he get $ F^{t}(p) = p $ for all $t$? Because if we set $\varphi(t)=F^t(p)$, it only has the conditions $\varphi(0)=p$ and $\varphi'(0) = X|_p=0$. This is a stupid question but I cannot understand.
Observe that $\varphi(t)=F^t(p)$ is the only integral curve of $X$ that at $t=0$ passes through $p$ with speed $X_p$. The curve $\alpha(t)=p$ $\forall t$ satisfies $\alpha(0)=p, \alpha'(0)=0=X_p$, so we just have to prove that $\alpha$ is indeed an integral curve of $X$.
For this, $\alpha'(t)=0=X(p)=X(\alpha(t))$, $\forall t$.
Since $\varphi$ is unique, then $\varphi(t)=\alpha(t)$.
$\textbf{Added}:$ So that you get a little more flavour from this. Suppose $p$ was an isolated zero of $X$. This is really the only case in which the local behaviour of a vector field is not almost completely determined locally by just knowing its value at a point. Since the trajectory through $p$ is constant, try to imagine how wild could the other close-enough trajectories be. Then look at p. 446 of Spivak's, Volume 1.