I think right now nobody knows the reasons why Klopp decided to leave Liverpool to go on break as he earlier stated as his reason for leaving Liverpool job and may be after resting for few months he is trying to reconsider his decision then Dortmund invited him to have some discussion who knows if it will end up seeing him as the new manager of Dortmund.
The management of Dortmund must have reached out to Klopp before he flew to Germany but I think if he end up accepting the offer to become the coach of Dortmund that means he left the Liverpool team because of the performance of the team last season as they couldn't win any trophy.
It could also simply be that the job at Liverpool was too stressful for him, because playing in the FA Cup in addition to the Champions League or UEL and on top of that having opponents like City or Arsenal in the Premiere League is just a different level of stress than playing in your own country in the Bundesliga where it is usually less complicated and you also have the factor with your own country where you usually prefer to stay, even if he likes Liverpool and the fans it is probably too much at some point.