What is the meaning of the statement:
In this case a map $(I^{i+1},∂I^{i+1},J^i) \to (B,A,x_0)$ is the same as a map $(I^i,∂I^i) \to (F_f, \gamma_0)$ where $\gamma_0$ is the constant path at $x_0$ and $F_f$ is the fiber of $E_f$ over $x_0$.
at the page $407$ in Hatcher's book?
For a map $\varphi:(I^{i+1}, \partial I^{i+1}, J^i) \to (B,A,x_0)$, define a map $\Gamma_\varphi: (I^i, \partial I^i)\to (F_f,\gamma_0)$ by $$(t_1,...,t_n)\mapsto (s \mapsto \varphi(t_1,...,t_n,s))$$ This map is continuous(see Proposition A.14). Now it is easy to check that the assignment $$\varphi\mapsto \Gamma_\varphi$$ is a 1-1 correspondence. Also, if you define a map $\pi_{i+1}(B,A,x_0) \to \pi_i (F_f, \gamma_0)$ by $[\varphi]\mapsto [\Gamma_\varphi]$, then it is straightforward to check that this map is a well-defined group isomorphism.