Dimensions of quotient rings and zeroes of ideals

Question

During the study of algebraic geometry (algebraic curves, to be more precise) I've found that there is some connections between the zero set of a given ideal and the dimension of some quotient rings (meaning the dimension of the vector space), i.e. for algebraically closed field $K$ it holds that the cardinality of $V(I)$ is not greater than the dimension of $K[x_{1}, ..., x_{n}]/I$ or that the multiplicity number of some point from the zero set equals the dimension of the quotient ring $M^{n}/M^{n+1}$, where $M$ is the set of non-invertible elements from the localization of this given point. Maybe these two examples have nothing in common but I do feel that there is some general philosophy in connecting zero sets with dimensions of the vector spaces. My question is how can someone come up with the idea that, for example, the zero set of $I$ and the dimension of $K[x_{1}, ..., x_{n}]/I$ do have some relationship to each other, or, more generally, that the zero sets of ideals can be connected to the dimensions of some quotient rings.