So you did see your father last night

This tender moment between Harry and Dumbledore opens a window into Harry's grief for his parents.

"It was stupid, thinking it was him," he muttered.  "I mean, I knew he was dead."

Harry is embarrassed to have believed he saw his father conjure the Patronus that saved his life.  He knows that the Mirror of Erised could not show the future, when Ron asks him if he thinks it does, because his family is dead.  And this is not the last time his logic will struggle to protect his heart, as he will see his parents again with Priori Incantatem in the Little Hangleton graveyard and with the Resurrection Stone in the Forbidden Forest.  It is not so difficult to imagine how the surprise and joy of learning he could leave the Dursleys and join the Wizarding World would wire Harry to believe, on some level, that other shockingly joyful realities could await him--such as being reunited with his parents.

Harry does not expect this to be possible, so he is embarrassed that he briefly considered the possibility, but rather than reminding Harry that reawakening the dead is impossible, he tactfully suggests that in a sense, Harry did see his father that night, in himself.  Harry's stag Patronus, newly corporeal, makes a more personal connection between Harry and his father than his looks alone ever could.  Without knowing what kind of animal his father became as an Animagus--only that it was large, and presumably had some connection to the nickname Prongs--Harry produces a stag Patronus.  The causality loop is confusing, as with any Time-Turner events, because Harry has already seen his Patronus and knows roughly what it will look like--a "galloping" animal, "as bright as a unicorn."

It is this combination of knowing and not knowing what his Patronus form will be that makes the eventual realization that it is a stag so satisfying.  Harry knows just enough, with his memory of the Patronus as he fainted and the limited information revealed by Sirius and Remus in the Shrieking Shack, and having seen his own Patronus already, he has the confidence necessary to produce it--and then he understands that his father's Animagus form must have been a stag.  Harry saves himself, Hermione, and Sirius from the dementors, but in a sense, his father saves them too.  Harry expects his father to save them, and at the last moment, he has to save them himself; knowing that his father would have saved them if he could gives Harry the strength to save himself, Hermione, and Sirius.