Matter cannot be created or destroyed. So when you apply that to regeneration, when every single cell is replaced, you have to wonder where the other cells go. When it comes to the Doctor, he exists so close to the edge of space and time, that with that final blast of regeneration (which is meant to push those cells out of existence to make way for the new ones), the cells are pushed past existence itself – through the Void, as it were. Experience has proven that one must be bound to existence to be contained in the Void. Case in point: the Dalek prison. Thus, while the Doctor all put together exists definitely, atomized and shoved over the edge of time and space, he cannot be bound. Therefore, he simply slips through.
But when he reaches the edge of the Void and begins to enter the next universe, that was when things with Nine became more interesting than with his previous regenerations. With those that entered the Void before him, they simply slipped through and then died on the other side. Their wounds did not disappear with reassembly. Their bodies simply reappeared in a world where no one knew them, and they were given the same treatment as any John Doe's dead body might. So regeneration, really, became the death of the Doctor in that state. Of his body, anyway.
Things with Nine, however, went a bit hinky. While his atoms were blasted away into inexistence like usual, the individual cells and their death were not quite the same. The Doctor had died of many things. He had never, however, died from Time. And, quite inconveniently (or conveniently, depending on how you looked at it), Time was one of those things that bound you to existence, and could have kept him stuck in the Void forever… had the Time been a part of his cells, instead of clinging to them. That tiny little fact was what saved him in the end. The Void held onto Time, as it does, causing it to vanish from everything. It was drawn from the cells (which didn't, in fact, exist), separating existence from nothingness, until his death was left in the Void, and he was left in sailing, inexistent Doctor bits.
He passed into the parallel universe, as all his previous forms had done, but left his death behind.
The universe requires that all cells account for themselves, and so if new cells that enter it are meant to be together, the universe will make it so. Thus, the unconscious (naked) body of a man was found floating past a quaint town along the Thames. It was not long before John Smith exhibited an extraordinary amount of medical (not to mention various other forms of) knowledge for someone with acute amnesia. How Tardis began to understand how this all happened is anyone's guess, but when her hypothesis had been properly formed, and her John Smith had shown the appropriate signs to verify it, she decided to work at his memory in earnest.