Well, although _there is a typo_ (Wiles forgot to close his parenthesis, and wrote $H^2(G,\mu_{p^r}\to H^2(G,\mu_{p^s})$ in his proof), his claim is correct. Let, as _ibid._ $F$ be the finite extension of $\mathbb{Q}_p$ fixed by $G$, so that your arrow can be written $H^2(F,\mu_{p^r})\to H^2(F,\mu_{p^s})$. Consider the Kummer sequence (for any $n\geq 1$)
$$
1\longrightarrow \mu_{p^n}\longrightarrow \overline{F}^\times\overset{(\cdot)^{ p^n}}{\longrightarrow}\overline{F}^\times\longrightarrow 1.
$$
Taking Galois cohomology and using Hilbert '90, which tells you $H^1(F,\overline{F}^\times)=1$, you find
$$
1\longrightarrow H^2(F,\mu_{p^n})\longrightarrow H^2(F,\overline{F}^\times)\overset{\cdot p^n}{\longrightarrow }H^2(F,\overline{F}^\times)
$$
Now, local class field theory tells you that the Brauer group $H^2(F,\overline{F}^\times)$ is isomorphic to $\mathbb{Q}/\mathbb{Z}$, so the above sequence identifies $H^2(F,\mu_{p^n})$ with the group $\mathbb{Z}/p^n$, seen as the kernel of multiplication by $p^n$ on $\mathbb{Q}/\mathbb{Z}$. If now you apply this for $r\leq s$, you see that the arrow you were firstly interested in is the injection $\mathbb{Z}/p^r\hookrightarrow \mathbb{Z}/p^s$ or, if you preferer, the injection
$$
\Big(\ker(\cdot p^r)\hookrightarrow \ker(\cdot p^s)\Big)\subseteq \mathbb{Q}/\mathbb{Z}.
$$