저야말로 잘 부탁드립니다. is a response to 잘 부탁드립니다.
저 is the humble form of 나 which is used to show respect to the listener.
-야말로 is a marker that adds emphasis on the preceding noun.
So, the conversation would go like this.
A: 만나서 반갑습니다. Nice to meet you.
B: 저도 반갑습니다. 앞으로 잘 부탁드립니다. Nice to meet you, too. I ask you to show me your kindness. (Ok, something like that. Well, you know what it means.)
A: 아니예요, 저야말로 잘 부탁드립니다. No, it is me who should ask your kindness.
I think the translation part( I ask you to show me your kindness) seems not explain the meaning well.
I would suggest to please be forgetful and nice when I do make mistakes and do something unpropriate.
And in response, the A person is saying, because the B person showed the humble side of him, it is him who should ask for such a favor. This way both try to put their position somewhat lower.