Proof:

There exists $g: Y \longrightarrow X$ such that $f\circ g$ and $g\circ f$ are respectively homotopic to id$_{Y}$ and id$_{X}$. By theorem 11.5 $f_*\circ g_*$ differs from the identity map on $\pi_1(Y, (f\circ g)(y_0))$ by a composition with the isomorphism $h_{[\sigma]}$ where $\sigma$ is a path joining $f(g(y_0))$ and $y_0$. In particular $f_*\circ g_*$ is bijective and so $f_*$ is surjective and $g_*$ is injective. Likewise, working with $g\circ f$ one concludes that $g_*$ is surjective and $f_*$ is injective. Hence $f_*$ is an isomorphism between $\pi_1(X, x_0)$ and $\pi_1(Y, f(x_0))$.