This is what I interpreted the movie to be **SPOILER**
These are my thoughts. They may be wrong, but here is what I am thinking. Please bear with me, I know this might sound strange and confusing, but I'm dealing with a confusing movie!
At the end or the movie, as Henry lay in the middle of the bridge, he has a Donnie Darko moment. Meaning, he imagines what his life would be like if he chose to live. I think that this is what we see in the first scene of the movie, when he walks away from the crash unharmed. Also when Henry is first introduced as Sam's patient, and we also see how Sam and Lila's lives are affected by Henry. From here on, to make it easier, I will refer to everything that happened before Henry's accident in the movie as a "Dream". Henry uses everyone he sees in front of him in his Dream, kind of like using them as the characters in his script.
As we see when he is almost dead, he feels like he is to blame for killing his parents, so in the Dream, he tells Sam that he killed his parents. Henry is depressed in his Dream, having lost both parents, believing he killed them, and wanting to kill himself to end his pain. I think he starts seeing the psychiatrist, as most suicidal young adults do, in the hopes of having someone care for them and help them. In the Dream, Henry is constantly looking for someone to save him.
When thinking about this idea, I then thought about his parents in the Dream and how they worked into this explanation. Then a thought occured to me. I believe Henry wanted to make ammends with his parents, so he made them real to be able to help/apologize to them. He helps his father by letting him see, like this could erase the fact that Henry killed him (so he thinks). I think he is imagining his mother as she always was, getting him something to eat, fussing over him, because maybe that's what he missed most, and he was looking for familiarity. I think that's also the reason why the dog was there.
I also think that when Henry and his mother began bleeding profusely, that it was a glimpse of Henry's current condition, outside of the Dream, as he lay dying on the bridge. All in all, in the end, he feels like there is nothing to live for and decides to kill himself in the Dream. That is where it morphed back to the present, where Henry decides it would be too painful to live without his parents, so he decides to die, this "killing himself" as he said he wanted to do.
I hope I have explained myself ok. I wonder if anyone agrees that this could be a possible explanation. If you have any questions you think I left unanswered, please feel free to reply, I would love to discuss. However, I ask that you please refrain from telling me I am wrong or stupid or that your explanation is correct. I don't believe that there is a correct explanation, unless it comes from the writer, and I have yet to see that. So please, respect everyone's opninions. Thanks!