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.
These could also be a good reason for him to have left the Liverpool team, the stress that Klopp encountered during last season is enough to make him to resign as the Liverpool manager because looking at what happened to him last season one could see that he became very tired because t some point the performance of his team became very poor when every one was expecting them to keep getting better and winning games since they were already in the race with Arsenal and Manchester City but the reverse was the case.
Coaching a team from his country won't be a bad thing if you ask me but for a manager that has gotten some level of respect he must continue to do well otherwise getting a sack letter if the Dortmund team fails to improve this season would definitely be shameful to him because staying in England as a Liverpool manager would give him an assurance to his job even though he doesn't win the league ahead of Manchester City and Arsenal.