What is "break in" and what difference does it make? In amps? Preamps? Speakers? More?

Yes, I know you never heard of it, sweet pea. That’s OK. Well, now you have. It’s all good. 🤗

Filter capacitors in amps and preamps have something called 'forming up' where they adjust to the operating voltages to which they are subjected, being electro-chemical devices. This takes some time and until that occurs the power supply will not be acting correctly. Not sure how long it takes, 50-250 hours? but its not that hard to measure that they do change- the power supply voltage itself goes slightly higher and there is less noise on the supply rails.

So that explains part of break-in. But coupling caps appear to have a break-in property as well and to me this isn't as well explained, but plenty of people report changes in coupling caps as they apparently break in.

We hear break-in artifacts in our own gear and for that reason (and to prove reliability) we play the gear in our shop before shipping. The first ten hours or so see the most dramatic changes and this period of break-in isn't experienced by most people as most manufacturers don't ship as soon as the bench tests are passed. We have noticed however that somewhere in the last 20 years that phenomena is vastly reduced over what we used to experience so we suspect that some part we are using is a bit better at its job now than 20 years ago. Generally speaking though its unimportant as we put at least three days of operation on something before shipment. Often we'll run it longer than that- up to a week. But customers consistently report break-in improvements and have been doing so as long as we've been in business, and having experienced it myself many times I just accept that its a thing. I think power supply filtration is most of it.

You can philosophize all you want, it just comes down to what you hear. I recently bought a tube headphone amp. It sounded nice, but a bit shrill. I started burning it in. Just 10 hours later, it sounded MARKEDLY better. Not subtle.

YMMV with other equipment, but I would ALWAYS break things in, just in case they need it. Some things definitely do!

Regardless of what we say, what we believe, or what we consider to be proven by the science of the day, does not change the truth. Believing by whatever basis however is real for us. In these small cases of what makes this or that sound better to us, is an important human reality. 

I suppose ignorance regarding the fact that interconnects ALL comprise an, "RLC" network, to some degee, is fairly widespread.      The, "C" is for, "CAPACITANCE" and- like any capacitor; the dielectric will take time to stabilize.      The higher the quality of dielectric, the longer the stabilization period.      I’ve always personally held that Dielectric Absorption has much to do with the way capacitors and cables sound.      Especially, given that the better the dielectric (ie: Teflon/Polypropylene/Polyethylene/etc), the lower the absorption figures will be.      https://www.sciencedirect.com/topics/engineering/dielectric-absorption      Of course, there are those that deny that science, with regards to the sound of a system.      https://www.elandcables.com/the-cable-lab/faqs/faq-what-is-capacitance