What does the axiom of replacement mean and why should I believe it?
On the Foundations of Mathematics mailing list some years ago, Arnon Avron argued that replacement is the way mathematicians naturally construct many sets. I quote one example from his article:
When asked to write a term denoting the set of singletons of elements of $\mathbb N$, I bet that at least 999 mathematicians (either in the broad sense, including first-year students, or in a narrower sense) out of 1000 would write: $$\{\{n\}: n\in {\mathbb N}\}$$ and not $$\{x\in P(P({\mathbb N})):\exists n\in {\mathbb N}. x=\{n\}\}.$$ This is not only because the former is shorter, but because it directly translates the definition in words of this set, and precisely reflects our intuition how this set is formed/constructed. In contrast, one has to think for a while in order to get the second definition correctly (and for many students it is even difficult at first to understand why this term is a correct description of this set. Anyone who have taught a basic course in set theory or discrete mathematics has experienced this). It is clear therefore that practically everyone relies on replacement for getting this set, and not on the powerset axiom.
According to this line of thinking, replacement is an intrinsic feature of any faithful description of the universe of sets, including the cumulative hierarchy.
For an argument that the iterative conception implies something weaker than unrestricted Separation (implied by unrestricted Replacement), i.e. $\Sigma_2$ Replacement, see Randall Holmes 2001 http://math.boisestate.edu/~holmes/holmes/sigma1slides.ps. (According to Professor Holmes, “this contain[s] an error, which Kanamori pointed out to me and which I know how to fix.”)
There is a wonderful blog post by Joel David Hamkins at Transfinite recursion as a fundamental principle in set theory which goes into great depth on this topic.
My answer to your question would then be "We should believe the axiom of replacement because we believe in recursion, even at the transfinite level."