TITLE: Is there an intuitive reason for Zariski's main theorem?
QUESTION [68 upvotes]: Zariski's main theorem has many guises, and so I will give you the freedom to pick the one that you find to be most intuitive. For the sake of completeness, I will put here one version:
Zariski's main theorem: Let $f:X\rightarrow Y$ be a quasi-finite separated birational map of varieties, where $Y$ is normal. Then $f$ is an open immersion.
There is no reason to pick this particular formulation. In fact, every formulation seems to me like a technical lemma rather than a theorem with geometrically intuitive content.
Question
Is there a formulation of Zariski's main theorem that has an intuitive/pictorial ``reason'' for it? Or is Zariski's main theorem in its core a technical result with no geometric reason?

REPLY [20 votes]: Reading these excellent answers made me realize I did not understand the ZMT in an intuitive way, and helped me very much get at least more intuition for it than I had.  I want to try to summarize what I have learned from thinking about the other answers here, in a way that is not too technical and hopefully intuitive.  I do not consider the proofs, but only the intuition contained in the statements.
The first step is necessarily a little technical since it involves the definition of normality.
1)  If X is a normal affine variety, i.e. if its affine ring has no non trivial module - finite extension within its quotient field, then it follows from this definition, that X accepts no non trivial finite birational morphism.  Thus X is normal if and only if every finite birational morphism Y→X is an isomorphism.
2)  ZMT then implies that every such normal variety is unibranch.  I.e. if X has more than one branch at any point, then X must accept a non trivial finite birational map Y→X.  The geometric intuition here is that it should be possible, by a finite morphism, to separate the branches of any variety that is not unibranch.  This may have been suggested to Zariski by examples such as the projection cited by Francesco.  Thus ZMT establishes that the only way for a variety to possess more than one branch is for it to be the target of a map similar to a “projection”.  (Mumford's nice topological and power series statements are just alternate ways to say "unibranch".)
3)  As Sandor pointed out, then the connectedness theorem follows naturally for a normal variety X, since if Y→X were finite, birational, and some fiber f^(-1)(p) were disconnected, then X should have at least two branches at p, hence X should not be normal.
4)  The next natural piece of intuition is Grothendieck’s theorem that all quasi-finite morphisms can be completed to finite ones, as Matt and Akhil observe.  Thus the only way to get a quasi - finite morphism is to restrict a finite morphism to an open set, a very intuitive geometric statement.  In particular, since there are by definition no non - trivial finite birational morphisms to X, there cannot be any non trivial quasi- finite birational ones either; i.e. every such morphism Y→X is an open immersion.
5)  As a consequence, a birational morphism Y→X which is not trivial, i.e. not an open immersion, cannot be quasi – finite, hence must have a positive dimensional fiber f^(-1)(p).
So for me at least, these various statements have all become geometrically natural, thanks to contemplating the other answers, admittedly in a naïve way.