Why do speakers improve with more powerful amps?

They don't.  Many will tell you they do but they were probably told that by a salseman with a powerful amp.  Just had a friend over last night and he was blown away by my 2 wpc amp.  You listen to the first watt.  If it sucks, you don't need 299 more of them.  

People have loved to brag about how many watts their system has for decades.  I did it in college. Now I know better.

I had a name brand 300 wpc, 120 lb amp and decided I needed new speakers. Put my speakers on the market and a guy came over to listen.  He brought his SET amp and we turned it on and I listeded 2 minutes and said, "I"ll honor my offer to sell these to you but if you don't buy they are off the market".  He didn't, maybe partially because he realized I wanted to keep them, and the big amp was sold within a week and with a quality SET the speakers shine.

2 nights ago I watched Guardians of the Galaxy with my son at theatre sound levels and my HT system is set up to  drive the front speakers with the SET.  The house shakes during battle or rocket scenes and the SET could go even louder.   You need 1 good watt with sensitive speakers, perhaps 5-50 good watts with less sensitive speakers.

"headroom" is also a marketing term.

I'm sure that there are many about to argue with me: think about it, unless you are listing at ear bleeding levels, your 300 watt amp is putting out .5 to 5 watts and I say that if you have a good 5 watts, then how can the other 295 which are not being used affect the sound?  

Jerry

With high efficiency speakers, one can compare high and low powered amps without unfair prejudice against the low-powered amp.  My preference tends to favor low-powered tube amps.  Tube amps tend to sound more alive and engaging at lower volume levels than solid state.  I don’t like most high-powered tube amps because the sound tends to be a bit harsh.  
 

My speakers are about 99 db/w efficient.  The amp I am currently running puts out about 5.5 wpc.  That is enough for me.

@carlsbad2 

+1!!! My listening experiences and progression through the years mirror what you have described. Without question it depends upon the specific speakers utilized. Certain designs do require/ benefit from higher power output and current capability.

If the speaker is appropriate I find that I prefer the presentation and sound quality provided with high quality low power amplifiers (Particularly DHT tubes).  There are numerous ways to achieve very satisfying sound quality via one’s audio system. Higher powered amplifiers as the better option by default is not a position I agree with. It depends on the specifics of the circumstances one faces.

Charles