why do expensive amplifiers produce a better soundstage

There is no correlation between amp cost and sound stage. Period. 

@toddalin 

I’m sure your system sounds great and I would love to hear it. If you are ever in Colorado you are most welcome to hear my system too! Happy listening !

From Nelson Pass: "...there is a tendency to interpret negative phase 2nd [harmonic] as giving a deeper soundstage and improved localization [of images] than otherwise. Positive phase seems to put the instruments and vocals closer and a little more in-your-face with enhanced detail."https://www.stereophile.com/content/gramophone-dreams-26-nelson-pass-harmonic-distortion

Second harmonic distortion is something he adds to his amps, which are expensive, so there you are.

Tubes also have it, but "naturally".

Something else about equipment that can detract from soundstage and imaging is poor channel matching, and this can vary with both volume and frequency.

The louder channel will "pull" the image toward that side and you won't get a good center image.

What are the key parameters that define amplifier performance?  Most would agree that specified power output into a required load is always a primary consideration.  A good amplifier is expected to deliver rated power at low distortion across a broad frequency spectrum into the load imposed by the chosen speaker system.  What else?  Slew rate should be considered, higher is better because this tells us that the amplifier can follow rapid changes in power demand.  Following slew rate should be rapid decay because if the amplifier overshoots the transient requirement it will sound shrill.  We can tell these things by looking at square wave response.  Or, perhaps even better we can use our ears.  More expensive amps do these things better than cheap amps.  But none of these things bears on the question posed by the OP.  It is probable that all of us have heard very expensive systems, perhaps at shows or what not, that did not image well.  Conversely, as confirmed by others in this thread, it is entirely possible to put together a system with superb imaging that does not incorporate super expensive amplifiers.  It is important to understand what the contributing factors are for the various aspects to good sound that we all strive to achieve.  Again there are good a sufficient reasons why a better amp might be desirable.  Understanding what each component is contributing is important to achieving a well balanced and good sounding system overall.