Although you say you're not interested in the algebraic aspects of things, restating things in terms of abstract harmonic analysis on locally compact abelian (=lca) groups might make things clearer.
For any locally compact abelian group $A$ and $\widehat{A}$ the group of it's unitary characters. $\chi \mapsto \widehat{f}(\chi) = \int_{A} f(x)\chi(x)dx$ with $dx$ a fixed haar measure on $A$. there is a measure $d\chi$ on the lca $\widehat{A}$ dual to $dx$ satisfying
$$ f(e) = \int_{\widehat{A}} \widehat{f}(\chi) d\chi $$
$\mathbb{Q}_p^{\times}$ is your abelian group, under multiplication. It's Pontryagin dual is the space of unitary complex characters $ \widehat{\mathbb{Q}_p} =\{ \chi: \mathbb{Q}_p^{\times} \to \mathbb{T} \}$ with $\mathbb{T}$ the complex circle. For any smooth compactly supported function on $\mathbb{Q}_p^{\times}$ it's abstract Fourier transform is the function $\widehat{f}(\chi) = \int_{\mathbb{Q}_p^{\times}} f(x) \chi(x) dx$
Because $\mathbb{Q}_p^{\times} = \mathbb{Z}_p^{\times} \times p^{\mathbb{Z}}$ it's unitary dual is $\widehat{\mathbb{Z}_p^{\times}} \times \widehat{\mathbb{Z}}$. The unitary dual of $\mathbb{Z}$ is $\mathbb{T}$. The complex parameter refers to the unitary character determined by $p \mapsto |p|_{p}^s=p^{-s}$ (or however you want to normalize it, but this is natural for number theory). Call this unitary character $\chi_s$. The remark about the unramified characters means it is a characters of $\mathbb{Q}_p^{\times}$ that is invariant under $\mathbb{Z}_p^{\times}$. Therefore you may think of the function "$\frac{1}{1-p^{-s}}$" more explicitly as a function $L$ on the unramified unitary dual
$$\chi_s \mapsto L(\chi_s) = \frac{1}{1-\chi_s(p)}$$ if $\chi_s$ unramified. $$ \eta \mapsto L(\eta) = 0 $$ $\eta$ not unramified.
The dual measure is always a haar measure on $\mathbb{T}$, which in this case looks like $\frac{\mathrm{ln}(p)}{2\pi i} \frac{ds}{s}$ (because the contour is over a parameter between $\frac{-\pi}{\mathrm{ln}(p)} $ and $\frac{\pi}{\mathrm{ln}(p)}$.
Finally we can write $L(\chi_s) = 1 + \chi(p) + \chi(p^2) + \cdots = \chi(e) + l_p \chi(e) + l_{p^2} \chi (e) + \cdots$ where $l_x$ is translation by $x$.
Inverting $ \chi(e) = 1$ gives a function $ f_{0}(e) = \mathrm{ln}(p)/ 2\pi i \int_{\mathbb{T}} 1 ds/s$ just gives you the measure of the circle = $1$. Because $\chi_s(\mathbb{Z}_p^{\times})=1$, $f$ must be invariant under and supported in $\mathbb{Z}_p^{\times})$
Similarly inverting $\chi \mapsto l_p(\chi)(e)$ gives you the same thing, a function $f_{1}$ such that $l_p(f)(\mathbb{Z}_p^{\times}) =1$ and is invariant under $\mathbb{Z}_p^{\times}$. In other words, this is the characteristic function of the coset $p \mathbb{Z}_p^{\times}$. Thus $ \sum_n f_n = 1_{\mathbb{Z}_p}$