Using the Intermediate Value Theorem on $h(x)=f(x)-x$ and $r(x)=g(x)-x$ I can easily show that $\exists t_f,t_g\in[a,b]$ so that $f(t_f)=t_f$ and $g(t_g)=t_g$. It remains to show that $t_f=t_g$. Since $f$ is 1-1 and continuous, $f$ is monotone.
Case 1: $f$ is strictly decreasing. If $g(t_f)>t_f$ then $f(g(t_f))<f(t_f)=t_f\Rightarrow g(f(t_f))<t_f\Rightarrow g(t_f)<t_f$ which is a contradiction. Similarly $g(t_f)<t_f$ leads to a contradiction and so $g(t_f)=t_f$.
Case 2:$f$ is strictly increasing. Here is where I am stuck :( Any hints?