The character is explaining that he got requested by an actress to join a talk show.
なんでも前に僕が出たテレビ番組を見て指名してきたんだそうだ
Is basically "I am told she saw the show I was on and nominated me."
年齢も近いし
"We are also close in age"
ぜひにって
Some googling suggest is could be ぜひにって言った so he's saying it "So I said ぜひに" basically saying he'll be there but I'm not sure.
