The first 'softening' in the British stance was the lifting of the broadcasting ban on Sinn Fein.*
Gerry adams also had a tour of the US.
In 94/95 we saw the rise to prominence in Ulster of the moderate Catholic John Hume - I think this reflected a sea-change in Catholic opinion, and I think Sinn Fein realized this.
What caused the change? The relentless violence on both sides of the aisle. The majority were always sick of it but they started voting for moderates like Hume and pressuring hard-liners like David Trimble into becoming more moderate. Both the British and Irish governments saw the need to stop the bleeding.

As much as I hated it at the time, Clinton stuck his oar in. I thought it was typical Yank meddling, but at the same time Britain and Sinn Fein started behind the scenes talks. Was this Bill seeing an opportunity - recognising the sea change - or did he start the process by encouraging both sides to talk? Dunno the answer to that. (I do think that without american funding via NORAID, the IRA would probably dried up years before, but that's another story)
Whatever it was, it worked, culminating in the George Mitchell report in late 95.

While idiots like Iain Paisley continued to chant "No Surrender", David Trimble also saw the opportunity and worked with Hume.
There followed a lengthy process, but I think by 96 it was really all over bar the shouting (and there was a lot of shouting).

So, IMO, Clinton played a large role in cementing the need for talks. Gerry Adams played a big role in co-ercing the IRA to end the struggle. Tribmle and Hume were instrumental in getting their sides on board.
British and Irish governments both put their weight behind the process and, for the first time, together were on the same page and paid more than lip service to the need for peace.

Compromises were made on all sides along the way - Sinn Fein was given the hope that someday they may be able to engineer a political return to Ireland. Unionists were given hope that they would remain British until the people no longer wanted to. With the Good Friday accords, both groups were assured at least some power.

There is still trouble in getting the whole thing to work, but certain paramilitary activities notwithstanding, the situation is a whole lot better now than it was in the seventies and eighties.

DFM may have been referring to the Stevens 3 report?
If so, while there was collusion between parts of the British government and paramilitaries, his assertion is still 100% wrong.........

*This was a ludicrous Thatcher ban. The British broadcasters were banned from airing Sinn Fein's (read Gerry Adams) speeches. They got around this by showing the video of speeches/statements with an overdub.
IMHO, lifting the ban was a PR disaster for Sinn Fein since the guy they used to dub had a rich Irish brogue, compared to Gerry Adam's piercing whine. :-)