There is a quotation of Stinespring dilation in a book about C*-algebra.
(Stinespring dilation) Let $A$ be a unital C*-algebra and $\phi: A \rightarrow B(H)$ be a completely positive map. Then, there exist a Hilbert space $H_{1}$, and a *-representation $\pi: A \rightarrow B(H_{1})$ and operator $V:H \rightarrow H_{1}$ such that $\phi(.)=V^{\ast}\pi(.)V$. In particular, $||\phi||=||V||^{2}=||\phi(1)||$.(which, applied to $\phi_{n}$, implies $||\phi_{n}||=||\phi(1)||$ as well).
I do not know how to apply $||\phi||=||V||^{2}=||\phi(1)||$ to $\phi_{n}$ here. Could someone explain this to me ? (Here, $\phi_{n}: M_{n}(A) \rightarrow M_{n}(B(H))$.)
What the paragraph says is that the norm of a completely positive map on a unital algebra is its value at the identity: $\|\phi\|=\|\phi(1)\|$. Now you can apply this to the completely positive map $\phi_n$: so $$ \|\phi_n\|=\|\phi_n(1_n)\|=\left\|\begin{bmatrix}\phi(1)&0&\cdots\\ 0&\phi(1)&0&\cdots\\ &&\ddots\\ \end{bmatrix}\right\|=\|\phi(1)\|. $$