Is there some intuitive explanation why Dedekind zeta functions contain so much information about their number field?
For example the residue at the pole $s=1$ relates several invariants of the number field. the class number formula.
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This question is a major open problem. One theoretical framework has been proposed that would explain why all the various zeta and L-functions (the ones attached to number fields, function fields, modular forms, algebraic varieties, Hecke characters, ...) have the same nice properties (analytic continuation, special values that "contain so much information," ...). It is known as the theory of motives. It is somewhat stymied by the fact that no one can (with proof) define the right category of motives, nor has anyone been able to for some 40 years. Milne's lecture notes "What is a motive?" are a great introduction to the theory; http://www.jmilne.org/math/xnotes/mot.html
For a particular answer to your question, the so-called Equivariant Tamagawa Number Conjecture in the theory of motives encompasses every one of the "information-packed" formulas we have about special values of L-functions. Of course, no one expects a proof any time soon (since such a proof would at once prove long outstanding questions about the category of motives, Birch and Swinnerton-Dyer's conjecture and the Stark conjectures).
Here's one more concrete idea. One important gadget in algebraic geometry is etale cohomology. Most L-functions that we know of can be defined in terms of the eigenvalues of various Frobenius maps (global and local) acting on etale cohomology groups. In many cases, by clever arguments using analogues of classical Betti cohomology theorems (Poincare duality, Lefschetz fixed point theorems, ...), we can express the results on special values of L-functions in terms of etale cohomology. In other words, the conceptual reason that L-functions defined from different objects behave so similarly to one another is that all these objects are algebro-geometric, and etale cohomology ought to be an awful lot like Betti cohomology. Of course, this is more believable for a variety over the complexes than it is for Spec Z, but it's a start.
This isn't an answer so much as a general comment that might help to reduce your surprise: a number that is a product of several important numbers actually encodes less information than any of those numbers individually. All you get out of it is that if you can compute all of them except one, then you know the last one. In other words you shouldn't think of the zeta function as "knowing" all of those invariants.
The class number formula is a very good example of the principle that the Zeta and L functions associated to a number field contain a great deal of information about the 'arithmetic' of the field. In general though, it is still a mystery exactly what information these functions actually contain.
I think that it is safe to say that there are many more conjectures than theorems about the link between the analytic and algebraic objects that we study in number theory. I would point to Starks conjectures, The Brumer-Stark conjectures, the Coates-Sinnott conjectures, conjectures in Iwasawa theory about the mu-invariant, Leopoldt's conjecture, etc. . . all of these can be thought of as links between analysis and arithmetic that will illustrate how much the Zeta and L functions know.
Beyond this, one can even think about the Zeta function as something which is associated to the scheme Spec OK which is a 1 dimensional object and then try to generalize these conjectures to higher dimensional schemes over Z, to elliptic curves, etc.
Of course this is more of a list of the ways that we think that these functions are telling us things and not so much an explanation of why they know. . . I think a lot of people wish they knew that! Certainly the fact that the Zeta function is built out of information about the ideal structure of OK gives some explanation for why it would know something about the arithmetic. . . and I think this 'obvious' connection is clearly visible in the proof of the class number formula. As for the rest of the conjectures that I mentioned. . . I think the connection is much more mysterious than obvious.