why do expensive amplifiers produce a better soundstage

I agree that transducers have a greater impact than amplifiers-which have one job, to amplify the signal passed to it, but, of course, a quality designed and built component will allow the rest of the system to perform at its best. Price is not an exact indicator of quality, but unfortunately the old saying that you get what you pay for is generally, maybe not universally true in audio just like anything else.

With that said, the room has a great effect on soundstage and the spatial cues engineered into the recording dictate what is possible.

Correct me if I’m wrong but @yamaho asked about amplifiers and how a more expensive design might improve soundstage. It’s nice to hear other’s opinions on differences among amplifiers and we probably should just stick to the topic; not room acoustics, speakers, etc everything else besides what the OP specifically would like to know more about. There are countless threads about the benefits of properly setup rooms and speakers. @yamaho if you are genuinely interested in amplifier design check out Douglas Self. He has a solid grasp on the fundamentals.

Peace to all

OP   a better soundstage

The best amp (audio system) should reproduce the original sound (or closer to it). More expensive amp mean nothing in audio. I can listen my humble WT amp ultra (modded Wiim amp ultra) forever, but I can’t bear to hear all other audio systems (incl. $million systems) for 10 seconds. They sound just too bad. ** I can listen Carlos269’s California Dreamin’ which seems to be the only other audio system I can listen other than my systems.

OP’s "better soundstage" may be not the same as my better soundstage. Every ears hear differently. So, OP should define what his "better soundstage" is. Alex/Wavetouch audio 

The original sound is impossible to know, though. 
In my opinion, as long as your amp and speakers are adequate for the size of the room, you should be good to go.

vvvvvv6  The original sound is impossible to know, though. 

What I mean by "the original sound" is the sound of original recording played in the system. 

Ex.) Compare  Carlos269’s California Dreamin’  vs.  Original Diana Krall - California Dreamin’. 

I hear Carlos’ is closest to the original music than others below live recordings. Alex/Wavetouch audio

Diana Krall- California Dreaming in Hong Kong

Diana Krall California Dream’in

Diana Krall - California Dreaming