It means it was his responsibility, role or choice to offer sympathy.
He may have been a family member, friend colleague who should offer sympathy to someone. 'He is her cousin so it is really up to him to say he is sorry first'. Or it could be that the bereaved person was there in the room and no-one else went forward, they felt shy or annoyed at something or weren't sure what to do. However, this man decided he had to do it as others didn't, so it was 'up to him' to do the right thing. 'John saw her standing alone and sad, so he knew it was 'up to him' to say something kind'.
Other examples, Mary what do you want to eat? It's up to you John (your decision, or choice).
It's up to her how she lives her life. (choice).
It's up to the government now, only they can change things (decision).
There was no food, and as the only breadwinner it was up to her and her alone to find some. (responsibility)