The mission this piece came off was CRS-4. This mission was one of the first missions where recovery experiments were done.
After separating from the second stage and payload, this first stage was spun around, the engines re-lit to slow the rocket down and allow it to re-enter the atmosphere on one piece. It then fell through the atmosphere, before the engines re-lit a third time to slow it down, and it splashed down at slow speed. At this point, the stage would have fallen over, and the pressurized tanks would have burst.
A good run down of the landing attempts are on a page in the reddit spacex wiki at www.reddit.com/r/spacex/wiki/dev
They identified that this was from this stage by eliminating all others.
It wasn't from the rocket that exploded, because that rocket had grid fins added, and the location where these fins would have been is on this piece of debris. In addition, not enough time has passed for debris from that event to have reached the Isles of Scilly. This also rules out any recent launch, for the same two reasons. It would also be assumed that the destruction of CRS-7 would have caused more damage to this piece than we see.
Of the other launches, they ruled out early ones because the designs of the flag and logo didn't match, and that they would have been destroyed by reentry as you said. That left about 5.
They then compared images of the rockets on the launch pad with the images of this debris. The locations of items (lumps and bumps, basically) on the interstage only matched for one launcher: the launcher for CRS-4.
Once this was confirmed, it matched with a serial number located on the stage. Earlier this year, a piece of the fairing from another launch was found washed up in the Caribbean. It had a similar serial number on it. SpaceX quickly confirmed what launch it was from, and it was noted that the 'core number', that is, sort of the serial number for the whole rocket, was embedded in a particular place in that part's serial number. In the same place in this piece's serial number was the correct core number for CRS-4. So they had their answer. All we are waiting for is SpaceX's confirmation, which should come in the next few days.