The following page is from Lee's Introduction to Smooth Manifolds.
Theorem 9.24 says，$\forall p\in\partial M，t\mapsto \Phi(t,p)$ is an integral curve of $N$ starting at $p$. We denote this integral curve by $f_p$, then the domain of $f_p$ is $[0,c)$ with $c>0$ or $c=+\infty$. $\forall a\in (0,c)$, then $f_p(a)\in\operatorname{Int} M$, is the domain of the maximal integral curve of $N|_{\operatorname{Int}M} $ starting at $f_p(a)$ an open interval of the form $(-a,b)$ with $b>0$ or $b=+\infty$?
We assume the domain of the maximal integral curve $g$ of $N|_{\operatorname{Int}M} $ starting at $f_p(a)$ is an open interval $(d,b)$ with $d<-a$ and $b>0$ or $b=+\infty$. We choose a number $t$ with $d<t<-a$, and let $h=i\circ g|_{[t,0]}$, where $i: \operatorname{Int}M\to M$ is the inclusion map, then $h$ is a continuous map from a compact space to a Hausdorff space, so $h([t,0])$ is closed in $M$, but $h([t,0])\subset \operatorname{Int}M$ and $p\in\partial M$ is an accumulation point of $h([t,0])$, which is a contradiction.