The unreasonable effectiveness of Logic in Mathematics. Why? Inspired by this post I would like to ask for an explanation of a remark often attributed to David Kazhdan concerning the fruitfulness of applications of mathematical logic to domains in mathematics not directly related to logic as traditionally conceived, and comparing those to applications of physics in mathematics in the sense of providing new intuitions that are not ordinarily accessible to practicing mathematicians via their traditional training.  So there are really two separate questions here: (1) How valid is the claim of such effectiveness? and (2) How valid is the comparison of physics to logic in this sense?
 A: I will not dwell on "unreasonableness" because in my opinion, the word is being used in an emotional way to express appreciation and wonder, not to assert a factual claim that the precise amount of effectiveness quantitatively exceeds some rigorously defined threshold of reasonableness.  And de gustibus non est disputandum.
However, I think that the part about "providing new intuitions that are not ordinarily accessible to practicing mathematicians via their traditional training" is fairly easy to explain.  Many applications of logic to other areas of mathematics center around some kind of transfer principle.  One way to think about transfer principles is as follows: We are studying some area of mathematics, and we are able to formalize not just the mathematical objects themselves, but everything we can say about the objects.  That is, we are able to rigorously define a formal language that is able to capture (virtually) everything we want to say about the objects.  Then by analyzing the formal language, we are able to draw conclusions about some other domain that is not quite the same as our original domain, but to which the formal language applies equally well.
This kind of argument does indeed involve a type of abstraction that is different from "usual" mathematical argumentation, because instead of examining the mathematical objects themselves, we examine the language that we are using to talk about the objects.  Examining mathematical language is a natural thing to do when considering "meta-mathematical" questions such as consistency; after all, how else can you analyze the limits of mathematical reasoning other than by formalizing mathematical language?  But the part that surprises some people is that the move from studying mathematical objects to studying the language used to talk about the objects can yield concrete results about the mathematical objects themselves, and not just abstract meta-mathematical results.  Without detracting from the awe and joy that we feel when we contemplate mathematical beauty, I would submit that this should not really be any more surprising than the general fact that mathematical abstraction—at least, the right kind of abstraction—can yield concrete consequences.
As for the analogy to physics, I personally don't think it goes beyond the truism that a different perspective can yield new insights.  For the parallel to be more than that, I think we would have to argue that the use of physical intuition amounts to an unusual process of abstraction, and this does not seem plausible to me.
A: The most striking thing I have seen for many decades, and perhaps throughout the last 200+ years of "modern" mathematics, is the use of model theory by F. Loeser and others, and then Ngo, to prove a certain form of Langlands' notorious "Fundamental Lemma" by model-theoretic means, quasi-magically transferring a function-field version of the result (proven highly-non-trivially, but, still, more physically-conceptually, by algebraic-geometric methods) to the number field context. Amazing!
But/and I do not know of any other recent (last 30 years?) results, though I would not claim scholarship here. (The Ax-Kochen things are a bit older, and perhaps do not have the same impact...?)
(I am acquainted with David Kazhdan a little, but have not heard direct comments from him in such direction. In fact, given my acquaintance with his general mathematical operational style, I would tend to think that any comments from him in such direction might indeed refer to the relatively recent application of model theory to prove a form of The Fundamental Lemma.)
EDIT: so, yes, as suggested by @Matt F., this is indeed an example of some magical/unreasonable power of logic/model-theory in (the rest of) mathematics.
For that matter, the Robinson's non-standard analysis, especially as nicely packaged by E. Nelson, is pretty magical and explanatory, in a way that seemed impossible by "direct mathematics".
A: Since mathematics is powerful on the mathematical level, it is only natural that it is potentially even more so on meta-mathematical level. Thus embedding metamathematical constructions back into mathematics, as mathematics, should indeed be extra impressive. Thus all this is about mathematicians getting into the habit of applying meta to mathematics, the difficulty is partly psychological.

A similar situation is still about the theory of categories. It should be more common to apply definitions for functions to functors, etc.

