This is an economic unit. Building this grants you the ability to store more mass/energy so that when you go in the red you have more time to sort it out before your stores are depleted. Note that engineers, factories, and command units also store some mass and energy. Building storage units adjacent to power plants or Mass Extractors or Fabricators grants an adjacency bonus in the form of an output bonus and gives a slight increase to the storage capacity of the unit.
Energy Storage structures detonate on death. A recent patch substantially increased the amount of damage done and the AoE of the death weapon. The same patch made Energy storage structures more necessary as the ACU's Overcharge weapon now requires 1000 more energy than a player starts with, meaning an Energy Storage structure is required to enable the Overcharge ability.
While completely surrounding a power generator by energy storage units will increase its productivity by 50%, this isn't strictly recommended for any power generator (unless you want the storage in the first place). When used on a T3 power generator, it will increase productivity by 50%, but will cost marginally more than 50% of the resources, however were you to build four on one side, you would increase productivity by 12.5%. Were you then to build another T3 Generator on the other side, sharing the storage, you would have a net increase of 25% (or 625 energy extra), for only 480 mass. Considering that that's significantly cheaper than a T2 generator (which would cost 1200 mass) and provides a larger bonus to income, this might make it profitable to do. However, you must bear in mind that were one of the T3 power generators to get destroyed, it would destroy the energy storage units, whose explosions (combined with the generator's itself) would kill the other generator, making this a very, very fragile method of energy generation to use.