The known technical part of burn in for tubes is forming the cathode. It's the last step in production, and involves running current through the tube for usually 20 to 200 hours. Before the cathode is fully formed, its emission (peak current capability) is limited and consequently the bias will drift until the process is complete. There is likely to be some excess noise as well. Not all circuits (amplifiers, regulators, followers, etc.) are equally sensitive to this process but it always happens inside the tube.
It happens on a molecular level. The speed of formation depends on the impurities in the (usually nickel) substrate material; the high-purity metal causes the formation process to take longer but the resulting cathode will last much longer as well. If an inadequate job was done at the factory, then you will need more time at home. I have not found any reference to the old-stock question - does the cathode need re-forming after a few decades of rest, even if it was fully formed initially? Anecdotally, it is not uncommon for an old-stock tube to benefit from some burn in. Obviously the process is expensive, tying up tubes and equipment and costing electricity, so manufacturers are motivated to do the minimum necessary.
Of course there may be other, unknown mechanisms at play as well. We hear a lot of things that we can't yet explain!
