Why do speakers improve with more powerful amps?

@akg_ca has it right. 

@jumia ….

Why do more powerful amplifiers impact speaker sound quality in a favorable way?

They don’t. Not always anyway. Several things should be considered. In example:

1. Amplifier power rating vs. speaker sensitivity

2. How does an amplifier handle impedance dips that some speakers present as a challenge to the amplifier. Some speakers with nominal 4ohm impedance may dip below 2ohms and not all amplifiers can or will handle it gracefully.

3. The size of a listening room and listening levels preference

Quality vs. quantity of watts and a proper matching based on the speakers you’re driving and size of your room are the most important considerations.
Just like with anything in this hobby, synergy is key.

The OP has just been presented with the reality that opinions vary widely on this subject.  And, there are a ton of variables to consider -- the speakers involved, their sensitivity and the load they present; the listener's desired volume; the size of the room being used; the performance capabilities of the amp involved; and so on.

For my two systems, I've found that I don't need a lot of power. I use a 20 watt/ch Aegir in my main system that I prefer to the much higher power amps I previously had. In my sunroom system, I'm using a 10 watt/ch Gjallarhorn. Both sound wonderful and provide all the volume I desire with no sign of stress or lack of dynamics. And, no, I do not have high sensitivity speakers. But I'm also not a headbanger when it comes to volume.

We circle round to the main point -- figure out what you enjoy and go with it. Attempting to impress others or change their minds results in meaningless debate and frustration, both a waste of your time.

So why the hell do people buy inefficient speakers?

So many issues with powering them, and yet so many people with positive feelings about amplifiers putting out a quality 50 W.

It's just nuts, and the communication about why this is is wacky.

So why the hell do people buy inefficient speakers? 

I raised basically this question.

https://forum.audiogon.com/discussions/low-sensitivity-speakers-what-s-special-about-them