My guess (and it's only a guess) is that there's a second newspaper already pre-folded to the size and dimensions of the smallest torn piece and then concealed inside the original paper (probably secured with low-tack adhesive).
See how the newspaper he produces at the end has creases exactly where those folds would be despite him not folding the original paper in order to tear it.
I suspect that it's concealed just inside the back cover, somewhere in the rough area of that blue "What a Save!" ad.
When he opens the paper at the beginning, he never actually opens the back page and if you watch his hands when he's tearing the paper, he always holds on to that section quite firmly and tears slower and more carefully than elsewhere.
As Steve already pointed out, watch carefully when he shows off the torn pieces of paper and moves them to the front of the stack.
The first two come from the bottom and he waves them around to show them quite clearly.
He digs into the middle of the stack for the third and final piece and moves it straight to the front of the pile just before shaking it out.
That is probably the pre-folded second paper.
As for making the torn pieces of the original paper disappear - I suspect that there's a concealed pouch inside the second paper that they all slip into when it's shaken out in a particular way.
In fact, just gluing the bottom of two pages together would automatically create a sort of pouch for them to slide into.
But that's the bit where the very clever sleight of hand comes in to play.
Or of course, it could really be magic!
Very well done though and I quite enjoyed watching it.