It just doesn't make sense, does it? Breaking bonds requires energy, and yet by breaking the bond of ATP, you provide energy for other reactions to take place! How does that work?
Essentially the answer can either be simple or complicated. If you want the simple version, it is this - when ATP provides energy, it's not simply breaking a bond, the bond is hydrolysed. A water molecule has to come in to break this bond. Because the water molecule comes in, some bonds are formed as well as bond being broken. So it's not just a case of energy being required to break bonds; energy is also released by the bonds that are formed.
If you're happy with that, you're best off leaving it there because the following gets quite complicated, and it needs to go all the way back to thermodynamics and ΔG. Indeed, it's actually something I struggle to understand myself!
The image on the left shows the enthalpy of formation for certain chemicals. This means the amount of energy released as a result of creating all the bonds in the chemical. The same amount of energy will be required to break those bonds, so if -2982kJ/mol is produced by creating a chemical, then +2982kJ/mol will be required to break it up again. This provides us with a very easy method of calculating how much energy is produced during a particular reaction.
Lets imagine that instead of just breaking a couple of bonds, we break every single bond in the whole chemical, and then we make the products of the reaction up from scratch. So to start off with we have ATP and water. If the reaction were simply breaking up these chemicals (i.e. opposite of formation), then it would have an enthalpy change of 2982 + 287 = +3269kJ/mol. However, then we make the products, which would have an enthalpy change of -2000 + -1299 = -3299kJ/mol. So while 3269kJ/mol is requires to break the bonds, 3299kJ/mol is released upon making the products, giving a net enthalpy change of -30kJ/mol.
If you take into account the fact that numbers have been rounded, and also consider that free energy considers entropy also, the final result that is always quoted is the -30.5kJ/mol, or 7.3 kilocalories per mole.
Of course, that's not too complicated. At least, hopefully not; indeed, if it is, leave it there, it's about to get worse!
At first glance, the whole thing doesn't seem to make sense. OK, so I'm prepared to accept that some bonds are formed, and that overall the enthalpy change is -30.5kJ/mol. All the same, it seems a bit odd that so much energy is released in the reaction. Well, there are good explanations for this, and it comes down to a series of important factors.
First of all, consider the ATP molecule. As you can hopefully see from the images, there are negative charges on the phosphate groups. These will repel each other, and will want to be apart from each other; so if water comes and splits up the molecule, the negative charges will be apart from each other. As shown in the animation, at least 1 phosphate group comes away and moves its negative charge at the same time. Since this is such a favourable reaction (after all, it 'relieves the strain' of negative charges repelling each other), it will make the reaction favourable, and will result more energy released, or a more negative free energy change.
Next consider the phosphate group. It is amphoteric which means that it can both donate and accept protons. This is clear from its structure. If the solution it is in has a low pH (i.e. lots of hydrogen ions kicking about) then it doesn't need to offer any, and it will keep them for itself - i.e. there will be no negatively charged oxygens, only -OH groups. However, if the pH is high (i.e. very few hydrogen ions kicking about) then the phosphate will donate some to the solution, and will end up with no hydrogen ions, only negatively charged oxygens.
Physiological pH (i.e. the pH that exists in most cells in the body) is about 7.4 which is fairly neutral. This means that a phosphate group will exist with just one hydrogen ion attached, and the remaining oxygens all free.
In the hydrolysis of ATP, as shown in the animation at the top, there is a hydrogen ion left over at the end. This is because at physiological pH, the phosphate group only wants its one hydrogen atom. Since it has that, the other hydrogen atom just has to float about. Therefore, at physiological pH, the hydrolysis of ATP will produce 3 things, despite only starting with two. This means that the system has become more disordered (because 3 is more disordered than 2) and there has been an increase in entropy, which means that the free energy will become even more negative.
Other factors which are considered include the presence of magnesium ions in most cells; since they have a 2+ charge, they reduce the repulsion between the negative charges on the phosphate groups, which means that instead of such a large enthalpy change, it is left at 30.5kJ/mol.
Finally, however, consider that the whole system is an equilibrium. According to Le Chatelier's Principle, if there is a build up of a particular chemical involved, then the equilibrium will shift to oppose that change. Imagine that there was a build up of ATP; the equilibrium would shift to oppose that change, meaning that more ADP would be produced.
In the body, this is exactly what happens. All the processes of metabolism are set up to produce more and more ATP, which means the reaction will want to happen to produce more ADP from the ATP - put in basic terms, there will be more energy released from the ATP → ADP reaction, because it wants this to happen more.
When we consider the amount of energy produced by the hydrolysis of ATP, we consider all of these factors together. Although the textbooks will tend to quote 30.5kJ/mol or 7.3kcal/mol for the value of energy released upon hydrolysis, if you take into account the high concentration of ATP that accumulates in human cells, the energy released from hydrolysis of ATP may be given as about 49kJ/mol, or 11.7kcal.mol. Which is, obviously, a lot!!
Essentially, it's best to look beyond the bond enthalpies to explain why ATP releases so much energy upon hydrolysis. It's not simply a case of considering formation of different bonds, but of considering that the formation of the products is favourable - it creates a system that is much 'happier', which means the reaction will release energy.