Using induction to prove transition states are the same
We use induction on the length of $w$. Let this length be denoted by $|w|$

Base case: $|w|$ = 1

Immediate by the assumption that $\forall a∈Σ. δ(s_0,a)=δ(s_f,a)$

Induction hypothesis: $δ^*(s_0,w)=δ^*(s_f,w)$ for $|w| = k$ where $k \geq 1$

Inductive step: Assuming the induction hypothesis, prove that it works for $k+1$:

Let $w$ be a word s.t $|w| = k+1$. Clearly $w$ can be seen as two strings $w = w^{-1} \circ v$ where $\circ$ is concatenation of strings. 

As $|w^{-1}| = k$ we, by induction, get that $δ^*(s_0,w^{-1})=δ^*(s_f,w^{-1})$. As $x$ is a string of length $1$ we get by $\forall a∈Σ. δ(s_0,a)=δ(s_f,a)$ that $δ^*(s_0,w)=δ^*(s_f,w)$

The proof could be a bit more formal, but atleast the intuition should be clear.