"I met her" is just informing that you met someone, whilst "I got to meet her" ("meet", not "met", because you've already used the Past tense in "got") suggests that you'd been waiting to meet someone, and you have finally succeeded. So the first one has no emotional overtone, and the second one informs the converser that you had been been trying to do something before, and you've finally made it.

You met her. It means that you saw her and might have a talk with her. You got to meet her, it means you have to meet her for some reasons.

I met her > you have seen or know her before. I got to meet her > 2 meanings > 1) you at last have the chance to meet/ know her 2) you have to meet her ( because 'got' means' have') Example : I got/have to go . I got a new bike for my birthday