Debbie Moon at 2011-10-03 07:48:28:
We're aware from the moment Burt addresses Leonard by his first name - after Leonard has introduced himself as if they've never met before - that something's wrong with this situation. That makes us interested in their relationship - and in order to work out what that relationship is, we pay attention to the exposition.
So we have an immediate use for the information - it explains what's going on between these two people we're interested in.
It's also useful having a character who can't remember what he's told people before and can therefore repeat himself with impunity! (Though in fairness, I don't remember that being overused in the film...)