The only explanation is that it was the last remaining piece of soul a.k.a. the soul in Voldemort. It couldn't have been the piece inside Harry because that was destroyed (or killed) when Harry was hit by Voldemort's curse. Also when Harry comes to, it seems that Voldemort had also passed out or have been in tremendous pain because all the death eaters were around him and they sounded worried. For my last argument I give a quote, said by Harry, from when he and voldemort where in there final battle:

"It's your one last chance, it's all you've got left... I've seen what you'll be otherwise... be a man... try... try for some remorse..."

Now there are two hints here.

The first is "I've seen what you'll be otherwise..." This tells us that Harry has already seen what Harry will be because it is in the past tense. Also, if Voldemort doesn't repent Harry will kill him, therefore when Harry says this speech he knows that the only thing Voldemort will "be" is a soul going "on" (as dumbledore puts it) and since the connection between Harry and Voldemort is through souls it is highly likely that the thing Harry sees at kings cross is the visual representation of voldemorts soul.

The second is "be a man..." This tells us that Harry knows that voldemort is just a naive little boy on the inside (or as a soul). It is also no coincidence that the thing at kings cross is also a child.

Also when dumbledore tells Harry that they cannot help the thing he is giving Harry a subtle hint that he can help voldemort in the real world just not in this place.

I think that the reason is that the piece of soul that was inside Harry wasn't completely severed from the soul inside Voldemort (the connection is the connection between Harry and Voldemort) and for this reason the piece inside Harry pulled the soul inside Voldemort into Harry's mind just before it was destroyed.