When a couple get married, it's conventional in English for the minister or the judge or whoever is performing the wedding to say, "I now pronounce you man and wife" or "I now pronounce you husband and wife". That is, he is saying that he pronounces, that is, declares, that these two people are now married.
I'm not sure what was meant by "I now pronounce you Mary and Joe." It sounds like either a variation on the usual announcement of a marriage, or perhaps making some joke comparing these people to a married couple.
The other person saying, "I now pronounce me very impressed" is making a joke keying off of the standard marriage announcement. He's saying it about himself, so instead of "I now pronounce you ..." he says "I now pronounce me ..." And obviously instead of "husband and wife" he says "very impressed".
Both the original, "I now pronounce you husband and wife", and the joke, "I now pronounce me", are not conventional modern grammar. I don't know if that was accepted grammar at one time or if it was always a stylized way of speaking. A full grammatically correct sentence would be, "I know pronounce that you are husband and wife." But we all know what it means. Even if it wasn't an accepted formula I don't think it would be hard to figure it out.