Мне-то=мне. = to me.
-то just adds some emphasis.
But me...
myself...
As for me, I...
etc.
Мне-то они заплатили, а вот остальным сотрудникам не повезло.
They paid me, but other workers were not that happy...
Мне-то всё равно.
As for me, I don't care.
Sometimes -то used without clear 'opposition' to others, just as emphasis, and sometimes without any obvious function, especially in old-fashioned 'rural' speech.
BTW The same particle -то - coming from the demonstrative pronoun тот-та-то - serves in Bulgarian/Macedonian as a post-positive article: -ът/-ят, -та, -то. A definite one.