Joe, RFH is the Royal Festival Hall in London....a fairly large concert venue.
The Transatlantic Sessions are basically an annual get-together at a "retreat" somewhere in Scotland (location varies), orchestrated by Aly Bain and Jerry Douglas, where a whole bunch of (mainly) Scottish, Irish and American musicians spend time there making music, the sessions (and additional background) being filmed and used to produce a series of TV programmes (which unfortunately are only broadcast in Scotland, so the rest of us have to wait for the DVDs to come out).
There's a "House Band" (Aly Bain, Jerry Douglas, Russ Barenberg, Donal Lunny and Michael McGoldrick being the main cadre). Other invited artistes contribute their own songs and the house band sits in with them and provides additional accompaniment where appropriate.
The whole shebang usually then does a few live tour dates....I think Glasgow, Birmingham and London, where at times you have 16-17 musicians on stage. I remember a few years ago seeing them at RFH one night and then driving 200 miles from home the next night to see Russ Barenberg do a solo gig in the back of a pub, seating about 60. The contrast was wonderful!