Back when Wen Chao forced them to look in the cave with the Black Tortoise of Slaughter(Xuanwu of Slaughter), in the midst of chaos the Wen maidservant, Liangjiao or Jiao Jiao, was jealous that Wen Chao kept on paying attention the Mian Mian, a cultivator from the Lanling Jin sect so she wanted to brand her using a hot iron on Mian Mian's face but Wei Wuxian blocked it for her and ended up with the crest on his chest.
I believe LWJ got drunk one time a year or two(maybe?) after WWX died and he was missing WWX. Basically he asked his brother for a flute but didn’t accept it. He ended up digging in Gusu Lan Sect’s storage and found a brand iron that was confiscated from the Sunshot Campaign and when he woke up the next morning, there was a brand in the same spot as WWX’s. As for what the brand means, I think the brand is just a sun symbol which is the Wen clan’s symbol. Basically they branded people with a sun to show their power/authority
After finding out wwx has died, lwj started to search at the cloud recesses for a flute. His brother gave him the most precious flute he could find but lwj didn't accept it, claiming it was not the one he was looking for. He got drunk that night and found the iron with wen sect mark, that's how he got the brand on his chest
As for wwx, he was trying to save mianmian from getting the brand on her face back when they were all at the xuanwu cave
(sorry for my English)

ah, i see lee hwa applying the same philosophy as the great hanguang jun:
everyday means everyday.