The idea is that over time your headphones get more accustomed to the types of music you listen to and therefore are able to more precisely recreate those sounds after time.
Not sure about 'burning in with loud music over-night'. I've always thought it was to do with the types of music you listen to, rather than just playing anything loud.
The people who claim it makes a difference tend to have 100-200+ hours 'burn-in' time on their headphones. I do think it's true to an extent though. My old DT770s had a reasonable amount of wear in time and I'm confident in saying the sound did improve between when I first got them and when I got rid of them. When I got my DT990s, they did sound a little 'imprecise' (for lack of a better word) at the start, but after a week or two of use that improved.
It's very unlikely it will make any noticeable difference to a gaming headset. It's more for studio headphones.
I suppose it's like anything brand new. It often takes a little bit of time for things to become used to you, and you used to them - cars being a good example.
