LISA (long PDF) uses interferometry. This is a method that allows very accurate measurement of the difference between two lengths.
Basically, a laser beam is split. Each half of the beam travels a different path. At the end of the paths, both beams are combined again. A small difference in path lengths causes a phase difference in the beam. This becomes visible as an interference pattern. From this pattern, you can deduce the difference in path lengths.
Because visible light has a very short wavelength (on the order of 400 nm), tiny differences in path length show up as a visible interference pattern. In effect, the path length difference is magnified.

LISA (long PDF) uses interferometry. This is a method that allows very accurate measurement of the difference between two lengths.
Basically, a laser beam is split. Each half of the beam travels a different path. A small difference in path lengths causes a phase difference between the beams. Both beams are combined in a heterodyne detector, which produces an output signal proportional to the difference in either frequency or phase of the two beams (I'm not sure which is used in LISA yet).
The laser produces a frequency on the order of $10^{14} Hz$, a frequency difference of 0.1 Hz is easily measured and yields an accuracy of $10^{-15}$.

Paper that calculates the accuracy of this method. (long PDF)
Report on the underlying technology of LISA (even longer PDF)

Edit: my initial answer was incorrect.