Below is how I tried:
Let $p:(C,c_0)\rightarrow (X,x_0)$ be a covering map.
Let $[\gamma]\in \ker(p_*)$
Let $e_X,e_C$ be the constant loops at $x_0,c_0$ respectively.
Then $[e_X]=[p\circ \gamma]$.
Let $F$ be a path-homotopy between $e_X$ and $p\circ \gamma$.
Then, there exists a unique homotopy $G:I\times I\rightarrow C$ such that $p\circ G=F$ and $G(s,0)=c_0$, by homotopy path lifting theorem.
I have no idea how this applies that $p_*$ is injective.. Please help.. Why is $[\gamma]=[e_C]$?