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 for this suggestion.
こちらこそ literally means "Consequentely the same here". (It's the same for me, I return the thank.) Actually, "こちら" is a... direction. It means like "on my side". And can even mean "over there" in a different context.