Why does bond breaking in ATP release energy?
In most chemical reactions where bonds are broken, other bonds are formed. Take your example of the hydrolysis of ATP. A bond between two phosphate groups breaks, but one of the phosphate groups forms a new bond with the oxygen of water. Whether energy is released or taken up (whether the reaction is exothermic or endothermic) depends on the sum of the energies associated with breaking and making bonds in that reaction.


  How does bond formation release energy?


In the simplest case in the form of kinetic energy, heating up the reaction mixture. In the case of ATP hydrolysis in biological processes, the energy is sometimes converted into mechanical energy (muscle contraction), used to run pumps (transmission of signals in the nervous system), or other processes that would not go forward on their own.