To my way of thinking, the other answers are missing an important
element, a necessary feature for a mathematical tool or method to
be called "trick."
Namely, in order to be called a "trick," a method or technique
must involve artifice or misdirection of some kind. When we treat a mathematical
object as something that it isn't really or when we pretend that
something is other than it is in order to advance an argument (which is not to suggest that the mathematics is not correct), then we
are using trickery. When we solve a problem by placing our focus on something else, in which we aren't actually interested as such and which may even be silly in some way—a kind of misdirection—but by doing so we become successful in the original problem, then we are using trickery. When we replace a robust concept, in which we are really interested, with a modified version of it, perhaps even an absurd version of it, but which makes the argument work, then we are using trickery.
For example, with Craig's trick, we replace a formula $\varphi$ with
the conjunction with itself
$\varphi\wedge\varphi\wedge\dots\wedge\varphi$ repeated many times
over. The new assertion is just silly and we don't actually care about it as such, although of course it is logically equivalent to $\varphi$. How could it possibly help? The point is that we can use the new assertion to code some extra information into an axiomatization or presentation: the number of times it was repeated. By this artifice, we can deduce that every computably enumerable theory
has a computable set of axioms. The same idea works in many other contexts. For example, every c.e. presentable group has a computable presentation, by sufficiently repeating relations suitably in the presentation.
With Scott's trick, the issue to be solved is that the equivalence
class of an object forms a proper class, which can cause certain
problems, and so we replace that equivalence class with the set
of rank-minimal members of the class. If we think of this fake
equivalence class as the real thing, then everything works great!
This trick is surprisingly robust, and can be used to find small
canonical sets of representing structures in almost any situation.
For example, in ZFC there is a definable manner of choosing a
set of groups from each group isomorphism class: the
rank-minimal groups from that class. This is a trick, because we
don't really care much about that particular collection as such.
With Rosser's trick, we replace the concept of a theory $T$
proving a sentence $\sigma$, with: $T$ proves $\sigma$ by a proof
for which there is no shorter proof of $\neg\sigma$. When you
think of "proof" using this concept, then Gödel's
incompleteness theorem is improved to the Gödel-Rosser
theorem, where one can drop Gödel's extra hypotheses about
$\omega$-consistency. This is a trick, because we don't actually
want to think about "proof" using Rosser's concept, except that
it makes the argument work.
In many of the other tricks, we do something that seems a little absurd at first, misdirecting our attention from the original problem to this other thing, which may seem irrelevant at first, but when we follow it more fully it provides the answer we seek.
In each case, we replace the concepts or objects in which we are
truly interested by concepts or objects that we don't actually
care about as such and which in several cases are comical versions of the original, except that they make the argument work.