'Bother (+object) ! ' is an old-fashioned (and more polite) of saying 'Damn (+object)! ' when you are annoyed and frustrated about something. He's fed up with burgling so he says the equivalent of 'Damn burgling!' I think it is possibly a non-vulgar variation of 'B*gger'.

In this case it means I am sick of burgling and everything to do with it. The weather that day had taken a nasty turn means that the weather had been good or OK and then changed to be much worse than it had been. So it might have been sunny and then started to rain heavily. When the weather turns nasty, it means that it is raining heavily or there is a big storm.

In that phrase "bother" means "troublesome". It would be better to say "bothersome burgling" but the speak strange in The Hobbit.