Localization of the Integer Ring

Intuitively, you can think about localization as a kind of 'zooming in' process on the prime $p$. The ring $\mathbb{Z}_{(p)}$ is like the ring $p$, but it is relatively more '$p$-focused'. This can be intuited in several ways.

One possibility is that one can understand the information contained in a ring as being expressed in terms of obstructions. Every time an obstruction occurs, this is telling you some defect of the space it's defined on. When you localize $\mathbb{Z}$ at $(p)$, you get rid of a lot of the information (obstructions) coming from other points (primes) than $p$.

For example, the fact that $2$ is not invertible is an obstruction in $\mathbb{Z}$ which is, in some sense, coming from the point $(2)$. But, when you localize $\mathbb{Z}$ at a different point, say $(3)$, this obstruction disappears--$2$ is now invertible.

So, what's left when you localize at $p$ is a ring which has forgotten information at all other points except for $(p)$ itself, from which you can conclude that you have 'zoomed in', or that the ring $\mathbb{Z}_{(p)}$ is a ring of functions on a space 'more local' to $(p)$.

Of course, it's not 'super local' (they are only 'Zariski local'). There are still obstructions in $\mathbb{Z}$ coming from other primes, which are not 'fixed' (forgotten) by moving to $\mathbb{Z}_{(p)}$. For example, the existence of certain square roots, and more generally, higher order equations.

To get rid of these obstructions, and in some sense, 'zoom in further' on the prime $(p)$, one must do a process called completion of $\mathbb{Z}_{(p)}$ at $(p)$. This zooms in further, and essentially only remembers the 'differential data' of $\mathbb{Z}_{(p)}$ at $(p)$--the only obstructions are coming from differential obstructions. But, I digress.

Hopefully that gives you a bit of insight as to how and 'see' $\mathbb{Z}_{(p)}$.

If you're interested in more of this line of thinking, I would suggest looking into the basics of algebraic geometry.