How is こちらこそ used?


I don't understand the usage of こちらこそ in the following exchange:

A: はじめまして、たなか ゆうこです。よろしく おねがいします。  
B: たなかさん、こちらこそ よろしく おねがいします。

In the translation, I was told that it meant "it is I, not you". Could someone
explain this to me?




It's being used here as B being humble and saying that the pleasure of this
meeting is all theirs.

You know how sometimes, especially in cliched movies, there will be an
exchange like this?

> X: Thank you for all you have done for us.  
>  Y: No, sir. **It is I who should** thank _you_.

The feeling is kind of the same here. Upon their first meeting, their exchange
goes something like this:

> A: How do you do? I'm Yuuko Tanaka. Pleased to meet you.  
>  B: Mr. Tanaka, **the pleasure is all mine**.

Or, if we interpret it (perhaps overly) literally†:

> A: How do you do? I'm Yuuko Tanaka. I humbly request that our relations go
> favorably.  
>  B: Mr. Tanaka; **no, no, it is I who** makes this humble request.

† Thanks to [@sova](https://japanese.stackexchange.com/users/9542/sova) for
this suggestion.

