What does line impedance stabilization mean?
It is all in the name. A Line Impedance Stabilization Network does just that. AC lines can vary wildly depending on where you are, how your house/outlet is wired, nearby radio stations, etc. 

The LISN is basically a large low-pass filter that gives you a "quiet" AC signal to, as Jasen said, measure conducted emissions from a unit. Different standards require different LISNs. For example, the LISN used in CISPR22 is diferent than MIL-STD-461. The end goal is the same though. The LISN ensures that you will read similar conducted emissions levels at different test locations. 

When performing a conducted scan you connect your spectrum analyzer to a 50ohm impedance within the LISN. I am attaching an image I did for work not too long ago that shows the gain and phase across frequency. This shows the LISN used for CISPR11/22. R3 is the impedance you would measure across, R1 is where you would attach the unfiltered AC. You can see I'm doing the sweep from the "output" which connects to the unit.