Proof:

By the previous result it is immaterial which point $x_0$ is chosen as the base point. Assume that $X$ is star shaped with respect to $x_0.$ Let $\gamma :[0, 1] \longrightarrow X$ be a loop in $X$ based at $x_0.$ We shall prove $[\gamma]=[\varepsilon_{x_0}]=1$ by constructing a homotopy $F$ between $\gamma$ and the constant loop $\varepsilon_{x_0}$, namely

$$F(s,t)= (1-s)\gamma(t) + sx_0.$$

This makes sense because $X$ is star shaped with respect to $x_0$. Turning to the boundary conditions,

$$F(0,t) = \gamma(t),\;\; F(1,t)=\varepsilon_{x_0}$$

$$F(s,0) = (1-s)\gamma(0) + sx_0=(1-s)x_0 + sx_0=x_0$$

$$F(s,1) = (1-s)\gamma(1) + sx_0=(1-s)x_0 + sx_0=x_0.$$