Question: Sorry for the catchy title. I want to understand, what should one have to do to disprove the Church-Turing thesis? Somewhere I read it's mathematically impossible to do it! Why?

Turing, Rosser etc used different terms to
differentiate between: "what can be computed" and "what can be
computed by a Turing machine".

Turing's 1939 definition regarding this is:
"We shall use the expression "computable function" to mean a function
calculable by a machine, and we let "effectively calculable" refer to
the intuitive idea without particular identification with any one of
these definitions".

So, the Church-Turing thesis can be stated as follows:
Every effectively calculable function is a computable function.

So again, how will the proof look like if one disproves this conjecture?

Answer: 
There's a subtle point that I rarely see mentioned in these kinds of discussions and that I think deserves more attention.

Suppose, as Andrej suggests, someone builds a device which reliably computes a function $f$ that cannot be computed by any Turing machine. **How would we know that the machine is in fact computing $f$?**

Obviously, no finite number of input/output values would suffice to demonstrate that the machine is computing $f$ as opposed to some other Turing-computable function that agrees with $f$ on that finite set. Therefore, our belief that the machine is computing $f$ would have to be based on our *physical theories* of how the machine is operating. If you look at some of the concrete proposals for hypercomputers, you will find that, sure enough, what they do is to take some fancy cutting-edge physical theory and *extrapolate that theory to infinity*. O.K., fine, but now suppose we build the hypercomputer and ask it whether a Turing machine that searches for a contradiction in ZFC will ever halt. Suppose further that the hypercomputer replies, "No." What do we conclude? Do we conclude that the hypercomputer has "computed" the consistency of ZFC? How can we rule out the possibility that *ZFC is actually inconsistent and we have just performed an experiment that has falsified our physical theory?*

A crucial feature of Turing's definition is that its philosophical assumptions are very weak. It assumes, as of course it must, certain simple features of our everyday experience, such as the basic stability of the physical world, and the ability to perform *finite* operations in a reliable, repeatable, and verifiable manner. These things everyone accepts (outside of a philosophy classroom, that is!). Acceptance of a hypercomputer, however, seems to require us to accept an *infinite extrapolation* of a physical theory, and all our experience with physics has taught us not to be dogmatic about the validity of a theory in a regime that is far beyond what we can experimentally verify. For this reason, it seems highly unlikely to me that any kind of overwhelming consensus will ever develop that any specific hypercomputer is simply *computing* as opposed to *hypercomputing*, i.e., doing something that can be called "computing" only if you accept some controversial philosophical or physical assumptions about infinite extrapolations.

Another way to put it is that disproving the Church-Turing thesis would require not only building the device that Andrej describes, *but also proving to everybody's satisfaction that the device is performing as advertised.* While not inconceivable, this is a tall order. For today's computers, the finitary nature of computation means that if I don't believe the result of a particular computer's "computation," I can in principle carry out a finite sequence of steps in some totally different manner to check the result. This kind of "fallback" to common sense and finite verification is not available if we have doubts about a hypercomputer.
The Church-Turing thesis has been proved for all practical purposes.

<http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.146.5402>

Dershowitz and Gurevich, Bulletin of Symbolic Logic, 2008.

(This reference discusses the history of Church's and Turing's work, and argues for a separation between "Church's Thesis" and "Turing's Thesis" as distinct logical claims, then proves them both, within an intuitive axiomatization of computability.)