"Дело" here means "cause, problem, matter, keypoint". "Дело в том, что..." is something like "the matter is...".
No, it is not translated "my aunt took me to the tutor". It is "my aunt took me a tutor", here "репетитор" is in accusative. So, it is like "she took (hired) a tutor (for) me".
If you want to say "my aunt took me to the tutor", it would be "Моя тетя взяла меня к репетиторy". Note: it would be "меня" (accusative), not "мне" (dative). The preposition "в" is impossible with persons, since "в" means "into", "to the point inside something".