Gravitational waves cause space itself to stretch in one direction and simultaneously compress in a perpendicular direction. In LIGO, this causes one arm of the interferometer to get longer while the other gets shorter, then vice versa, back and forth as long as the wave is passing. The technical term for this motion is "Differential Arm" motion, or differential displacement, since the arms are simultaneously changing lengths in opposing ways.
As described above, as the lengths of the arms change, so too does the distance traveled by each laser beam. A beam in a shorter arm will return to the beam splitter before a beam in a longer arm--as the wave passes, each arm oscillates between being the shorter arm and the longer arm. When they arrive back at the beamsplitter (where they re-merge), the light waves no longer meet up nicely; they are out of phase. Instead, they shift in and out of alignment for as long as the wave is passing. In simple terms, this shifting alignment results in a flicker of light emerging from the interferometer.