Turn squares on and they will each show a number indicating how many adjacent squares are lit up. The goal is to have, as the chart shows, 1 square that's all alone, 3 squares that have one neighbor, 4 squares that have two neighbors, one square that has three neighbors, and one square that has all four.
A good place to start is with the one that has all four neighbors. So, make a plus-shape of lit squares somewhere. Then place a square in a far corner of the map (this will be the one that has no neighbors), and just try turning various other squares on and off from there.
There's only two functionally-different places the plus-shape can be. It can either be dead center, or one tile to the side. If you can't solve the puzzle with where you put the plus, try moving it to the other possible location.