I'm going to copy/paste a chunk from section 3.3 that explains some of this better than I could. If you have any questions beyond that I might be able to answer them.

The TOA lightning location technique is based on the computations of hyperbolic curves. The emitted radio signal of a lightning discharge is traveling through the air with the speed of light. This is approximately 300000 kilometers per second, or equivalently, 300 meters per microsecond. Each received signal gets a time stamp. Let t_A (s) be the time stamp for signal s from station A. Time stamp t_A (s) is the Coordinated Universal Time (abbreviated UTC) in microseconds with an accuracy of +/- 1 microsecond. The difference of two time stamps for the same signal received by two different stations and the positions of these two stations define a hyperbolic curve.

What I'd wager is happening is something along the lines of the following. Lightning strikes over the ocean and emits a radio signal at c (speed of light). This travels however far with a dissipating amplitude and reaches multiple receivers in different locations. These receivers have an amplifier that increase the signal to a usable level, records the data of the radio signal and whatever else it picks up. All receivers that pick up the given lightning strike from over the ocean have their data compiled and used to triangulate the location of that strike using the hyperbolic method that I quoted above.