How does releasing a phosphate group from atp provide energy for a cell?

ATP is a very unstable molecule, so it takes quite of energy to keep it together. The 3 phosphate groups repel each other since they are all negatively charged. By hydrolyzing the gamma phosphate, it releases some of that stored energy. It's like holding together 2 south poles of a magnet. If you let them separate, the energy that you were using to hold them together is released.