It is mentioned in Wiki that one cay show that the Krull dimension of a valuation ring $D$ is equal to the cardinality of the set of proper subrings of the fraction field $K$ containing $D$.
Moreover, it is also claimed that there is a bijective correspondence between $\operatorname{Spec}(D)$ and the set of all subrings in $K$ containing $D$. I can step by step follow the proof (mentioned in Wiki) but I do not have a good understanding why it is true. Is there any excellent explanation?