Usually in biology (and being ATP, it most probably is biology), it's one of two things.
The gamma-phosphate (the third one, the one farthest from the adenosine) is very unstable, meaning the phosphoanhydride bond is easy to break. The cell "allows" it to break, but only at the cost of moving the phosphate to some other molecule, such as a serine or glycerol or fructose or whatever. This phosphorylation creates a bond with lower energy than the phosphoanhydride, and so is overall favored. Imagine the personification: the gamma phosphate hates being attached to anything, but hates being attached to an ADP the most.
Alternatively, if ATP hydrolysis is coupled via an enzyme, it is usually done through transient storage of the energy is protein conformation. An enzyme binds ATP, which makes the protein structure "bend" or conform around the ATP. This puts loads of strain (energy = A) on the protein which is offset by the stabilization of binding the ATP (energy = B). This strain can make an enzymatic surface open up on the protein which itself takes a lot of energy to make (energy = C). The surface can catalyze some reaction (X+Y->Z in your example) that costs some energy (energy = D). The completion of that reaction alters the enzyme's catalytic site to something new and higher energy (energy = E), which can be alleviated by cleavage of the ATP (-7.3 kcal/mol). Alas, ADP and P do not fit well into that site of the enzyme, so they float out, restoring that original ATP-binding surface to it's original state. Provided A>B>C>D>E>-7.3, the cycle will continue until the ATP is exhausted or you have no more Z to make.
Typing "enzyme catalysis cycle ATP" gives a few examples. Here's a few: