Well, that sort of explains things then, doesn't it. The Tenth Doctor regenerates into the Eleventh Doctor. The Tenth Doctor's hand regenerates into the Tenth Doctor. Since "Hand of The Doctor" is tied into the Tardis every bit as much as the Doctor himself, the Doctor that grows from the loosed hand remembers everything that's happened over the past three seasons. And, indeed, the Doctor's entire life. Even though technically he's just a hand. Both face off against Davros. OMGWTFBBQ?! ensues.
Iirc, there was some kind of telepathic memory sharing thing the Doctor did with past incarnations of himself sometimes when he met. I think it was called "Contact" - they can easily use this to explain a second Doctor having his recent memories.