What is the intuition for $p$-adic numbers in wiki-sense?

For a locally compact abelian group $G$, one defines its Pontryagin dual group to be$$\widehat{G} = \text{Hom}_{\text{cont}}(G, S^1)$$where $S^1$ is the unit circle in $\mathbb{C}$. Thus an element of $\widehat{G}$ is a continuous homomorphism $f$ from $G$ to $S^1$. There is a natural map $G$ to $\widehat{\widehat{G}}$ (yes $\widehat{\widehat{G}}$, not $\widehat{G}$) that sends $g$ in $G$ to the homomorphism $\widehat{G}$ to $S^1$ sending each $f$ to $f(g)$. Pontryagin duality states that this map is an isomorphism, so that $\widehat{\phantom{G}}$ is an antiequivalence of categories.

The picture is a representation of the map $G$ to $\widehat{\widehat{G}}$ in the case $G = \mathbb{Z}_p$.

In more detail: The Pontryagin duality exchanges $\mathbb{Z}/n$ with the group $\mu_n$ of $n$th roots of unity in $\mathbb{C}$, and therefore $\mathbb{Z}_p$ with the group $\mu_{p^\infty}$ of all $p$-power roots of unity. So for selected elements of $\mathbb{Z}_p$ it is depicting a homomorphism $\mu_{p^\infty}$ to $S^1$. The elements of $\mu_{p^\infty}$ are the points of the flower-like designs, and the values in $S^1$ are the colors. One can see that $x$ and $y$ in $\mathbb{Z}_p$ are close exactly when their homomorphisms $\mu_{p^\infty}$ to $S^1$ agree on larger subgroups $\mu_{p^n}$.