See page 5 of the following Acrobat document:
http://www.schematicheaven.com/boogieamps/boogie_lonestar.pdf
For our purposes, I think this part of the schematic is similar, if not identical, for both the LSC and LSS.
Solo is a pot wired as a variable resistor (i.e. only the wiper (center) and one side are used). When Hard Bypass is off, Output is driven by the network off V3B (12AX7 tube in the effects return loop). There is a 100k resistor followed by a .68 uF cap, followed by another 220k resistor. This 220k ties to the top of the master output pot which ties to the top of the Solo pot.
If I'm reading this right (corrections welcome), then when Solo is off (normal condition), then the bottom end of the Output pot is tied to ground. If you turn Output all the way up, then the signal is effectively the output of V3B split between the 100k/.68uF/220k and the 1 meg pot. The 1 meg wins, so you get lots of signal. Turn Output all the way down, and it's at ground, so no signal to out.
When Solo is on, then, the value of the Solo variable resistor gets added onto the bottom of the output pot, thus increasing the resistance for the bottom part of the network and increasing the proportion of output signal sent to the driver stage. If you turn the Output pot all the way down, you'll still get signal unless you also turn Solo all the way down. Bump Solo up even a little bit and you should get some signal out. (I haven't tried this on my LSS but it should work that way.)
Now, this output network has a capacitor in it. While this looks like a simple high-pass filter, from doing a couple quick calcs, it looks to me like the sole purpose of that cap is to keep DC from being passed to the output stage. Once Output is turned up much, the cutoff frequency is already pretty low...a few hertz at most. Adding Solo has little effect on that cutoff frequency. And - since it is a high-pass filter - we're really just talking about taking out really low-frequency signals - DC.
I suspect what you're hearing in terms of "clearing things up" is simply that the output tubes are being pushed more, giving a bit more of the power tube distortion and energy that we like.
Again, corrections welcome. 20+ years since I have used this knowledge on a regular basis.
Dave