"Please don't smoke anymore!"
And this (Elementary use of A2 了): https://resources.allsetlearning.com/chinese/grammar/Uses_of_%22le%22
It's a set structure here: 别...了, 别再...了
再 can mean many things. In this case, it is used to imply that this person has been smoking more than once, and 再 means 'again'.
Without 再, the sentence would just be 请别抽烟了, translates to 'Please stop smoking already!' (already is added because of the 了, without the 了 and 再 it would just be 'Please stop smoking!') which doesn't make it obvious that the person has been smoking more than once, and could have just started.
再 can be also used in 'goodbye' (再见), because 'goodbye' is also 'see you again', the 再 being the 'again' in 'see you again'.
The sentence is trying to tell the person to stop smoking once and for all, therefore 'anymore' is used instead of 'again'.