why do expensive amplifiers produce a better soundstage

Assuming good room acoustics and speaker setup and just focusing on the immediate question:
-Higher quality power supplies.

-Superior noise rejection.
-Better topologies to take advantage of and work in conjunction with the lower noise power supplies.

It is a complex relationship within the entire amplifier design from power socket to speaker terminals, but for sure a common factor is very low dynamic noise floor. If the microdetail is blocked by higher uncorrelated noise the sense of air and space (soundstage) is diminished accordingly. The very finest amps do not mask low level detail, but get out of the way of the upstream signal, which also should be on a similar performance level of the amp to get the most out of the system. 

I use a 2003 Yamaha RX-Z9 RECEIVER in "Pure Direct" mode through my Merman speakers and have not heard anything that has ever "soundstaged" and "imaged" as well as my system in my room..., and I’ve been to CES and T.H.E. Show.  This is just the two speakers (18"/10"/AMT), and there is no eq or room correction, either electronic or physical being used. 

The system brings the venue into the room rather than taking the room to the venue. And the equipment has the volume and headroom to bring ANY SIZE venue into the room.  Total investment is <$6K.

https://youtu.be/OmWNOi6b4ak

It seems to me that if an amplifier contributes to soundstage and imaging, it is through improved channel separation and crosstalk rejection.

.,,,The Room‼️and speaker placement…. 

here is the answer to your question 

it comes down to parts quality an expensive amplifier uses higher quality better tolerance parts if you use cheap resistors they may not be of the same level they are reported to be same with capacitors enough variation in the circuit leads to loss of signal 

What is expensive?

$2k

$5K

$10K

$25K +++++++++++++++++++++++++++++++++

Those who are buying upper tier equipment also understand room acoustics to a degree.