Proof:

Let ${\tilde\gamma}$ be a loop in ${\tilde X}$ based at ${\tilde x}_0$ that represents an element of ker$\;p_*$. This means the loop $\gamma = p\circ {\tilde \gamma}$ is homotopic to the constant loop in $X$ based at $x_0$. But the constant loop $\varepsilon_{x_0}$ at $x_0$ lifts as the constant loop $\varepsilon{\tilde x_0}$ at ${\tilde x}_0 \in {\tilde X}$. By the covering homotopy theorem we conclude that ${\tilde\gamma}$ and the constant loop $\varepsilon_{{\tilde x}_0}$ are homotopic. That is to say $[\tilde \gamma]$ is the trivial element in $\pi_1({\tilde X}, {\tilde x}_0)$. $\square$