I am trouble understanding how one obtains a $\Bbb Z_p$ action in the last line in this statement in pg. 36 of Serre's Local fields
In particular
Observe that $\Bbb Z$ injects into $A$ and by continuity $\Bbb Z_p$ injects into $A$.
Maysome one elaborate on the details?
The injection $f:\mathbb{Z}\to A$ can be extended to $\mathbb{Z}_p$ by writing each element of $\mathbb{Z}_p$ as a limit of elements of $\mathbb{Z}$, using the completeness of $A$. Specifically, given $x\in\mathbb{Z}_p$, choose a sequence $(x_n)$ of integers converging to $x$ in the $p$-adic topology. Note then that the sequence $(f(x_n))$ is Cauchy with respect to the valuation topology of $A$: if $m$ and $n$ are large, then $x_n-x_m$ is divisible by a large power of $p$, which means $f(x_n)-f(x_m)$ has large valuation since $v(p)\geq 1$. So by completeness of $A$, $(f(x_n))$ converges to an element of $A$ we can define as $f(x)$. It is easy to see that this element is actually independent of the sequence chosen (the difference between any two such sequences is eventually divisible by arbitrarily large powers of $p$, so their images under $f$ are getting close in $A$). Similarly, it is easy to see that the extension of $f$ to $\mathbb{Z}_p\to A$ defined in this way is a homomorphism. Finally, the extension is injective because every nonzero ideal in $\mathbb{Z}_p$ is generated by some power of $p$, but the kernel cannot contain any power of $p$ since the original $f:\mathbb{Z}\to A$ was injective.
(What does this have to do with "continuity"? Well, $f:\mathbb{Z}\to A$ is continuous with respect to the $p$-adic topology on $\mathbb{Z}$, again because of the assumption that $v(p)\geq 1$, so this extension is just the unique way to extend $f$ continuously to all of $\mathbb{Z}_p$.)