Recall that a minimal prime ideal p lying over an ideal Iis the minimal among all prime ideals containing I. That is, if I⊆q⊆p, then q=Ior q=p.
In our case, we have that R is a PID. We are trying to show that all prime ideals are maximal. Consider a prime ideal p⊆R. It is a principal ideal since R is a PID. It is also a minimal prime ideal since it contains itself. Thus by Krull's PID theorem, has height at most one.
If the prime ideal is the zero ideal ( p=0), then it has height zero.
If it is any other prime ideal (p=(0)), then it has height at least 1, since there is the chain (0)⊊p. Thus by Krull's PID theorem, it has height exactly one.
So all the prime ideals other than the zero ideal, that is, all the points of Spec(R) have height 1.
Thus, every point of Spec(R) is maximal, as there are no "higher points" that cover them.