The basic idea is that you can only hold so many chunks in working memory (which is a necessary precursor to long-term memory), but the more you know, the better you can chunk information, thus remembering more.

There is a similar phenomenon in chess where grandmasters can very quickly memorize a board position from a game, but are basically no better than any random person off the street at memorizing the position of pieces placed randomly in a way that would never occur in a game. They aren't actually memorizing the position of pieces, they're memorizing the game state in a way that makes sense to them and lets them re-create the position of the pieces.

So, practically, a good way to train your memory is to learn more about the area in which you want to be able to remember things.

I don't think this helps with the concrete example of "something someone told me in a conversation" though. It might help with "something I read from a book" though because you could encode that information better if you know more about the subject.