Understanding when it drops below the LCL provides two indications in my experience. First, as mentioned, it might indicate that things have improved; in those cases its good to be able to identify what has changed to drive down the defect rate so it can be permanent captured as part of the process improvement. On the other hand, it is also possible that the number of defects identified is also driven by the inspection system effectiveness; in those cases the drop is actually a bad thing because your inspection is failing to identify the nonconformances. In those situations, you want to perform root cause and correctice action to the inspection system.
Either way, the LCL provides an indication that something has changed - root cause analysis will tell you if it is a good change or a bad change and you can then take corrective action accordingly (to fix it or make it permanent).
David