How is Kleene's T predicate defined? What I don't understand is how to extract information from the number that encode the computation history.
I know it's defined in Kleene's Introduction to Metamathematics. But what page?
References are welcome. More information on Kleene's T predicate can be found here.

There is also a more indirect proof method, which is to first show that any computable relation is definable in arithmetic and then note that the relation represented by the T predicate is computable. In fact the relation is primitive recursive, but the computability is easier to see, via Church's thesis. Also, not only the computable relations are definable, all the arithmetical relations are also definable. 
The proof that every primitive recursive relation (or function) is representable in arithmetic is somewhat easier than a proof specifically for the T predicate, because you can ignore details of Turing machines. Essentially the only issue is proving that one can quantify over finite sequences. There is a complete proof in many textbooks and in section 2.2 of these lecture notes by Stephen Simpson: http://www.math.psu.edu/simpson/notes/fom.pdf . 
There is also a proof in section 49 of Kleene's Introduction to metamathematics, at least for primitive recursive functions. The general result for arithmetical relations follows immediately by simply adding quantifiers and using the normal form theorem from computability to show that one-quantifier relations are definable.  This may require proving that the T predicate is primitive recursive, but this is easier than proving it is representable in arithmetic, because one can use primitive recursion freely without having to worry about the $\beta$ function at the same time. 
