What is the mechanism behind sympathic effusion of serous cavities due to nearby abscess?
The left perinephric space is adjacent to the left subpleural space so inflammation can spread from the former to the latter.

Diaphragmatic inflammation then results in a left exudative sterile pleural effusion. It's called sympathetic since it's not infective.

https://radiopaedia.org/articles/perirenal-space

https://emedicine.medscape.com/article/1979032-clinical#b4