FSD supercharging is the process of flying into a high-energy relativistic jet of a neutron star or a white dwarf in supercruise, which increases the FSD range for the next jump at the cost of slight damage to the Frame Shift Drive itself. FSD supercharging from a white dwarf gives a 50% range bonus, whilst doing the same with a neutron star provides a 300% bonus, effectively quadrupling your range for a single jump. Note that supercharging is not an instant process, and the pilots may have to spend several seconds inside the beam with a Fuel Scoop for the effect to apply.
However this is a risky process, as not only your ship's controls would be disrupted by the high-energy beam of the star, but disengaging supercruise while inside the jet will result in the entire ship experiencing major strain to all its modules and the hull itself. So far it seems like bigger ships such as Asp Explorer or Anaconda are much more resistant to those effects than smaller ones, e.g. Eagle.