Purely by piece counting, B+N is worth R + 1P. But I've seen lots of time when the exchange favours the side who holds B+N. In one variation of Scotch, white exchanges R+2P for B+N and is still slight in favour.
What are the ideas surrounding this type of exchange?
Only a general answer can be given for this:
Generally the exchange of R+P (sometimes an extra pawn) for a Bishop and Knight favors the player who has the Bishop and Knight afterwards due to the simple fact that the Minor Pieces are much stronger during the middlegame than the Rook. You have two attacking pieces vs. one attacking piece. The side with the Bishop and Knight will have better attacking chances as they will have more open lines (diagonals are much more open than files in the middlegame) and good squares for the Knight. The Rooks will be blocked by the pawns and unable to contribute much to the game.
Of course, it is very position dependent. In various variations of the Sicilian Defense, Black will exchange a Rook for a Knight on c3, sacrificing the exchange. This is extremely important for Black to do as it creates weaknesses and increases the strength of their dark squared Bishop.
In addition to this, there are many attacks that use the Rooks in the middlegame by opening up the h-file. It's very position dependent as to which group of pieces is more important or not.
As a general rule of thumb though, a Bishop + Knight is considered more important and stronger than a Rook + Pawn.
As a general rule of thumb I would say
Knight + Bishop = Rook + 2 pawns
Following factors influence evaluation of this material imbalance
I think, in a middle game scenario when there are more number of pieces and pawns on the board, giving up a single Rook for two pieces (Knight and Bishop) is a good trade. This might be because the Rook does not enjoy free movement when there are no open files and too many pieces/pawns on the board in the middle game. As always the statement of cliche in chess "Depends on the context".
Chess is a dynamic game and so the pieces get their values according to the position on the board at the time and what they are doing. This is fluid and subject to change.
That said, as a general rule the rook side of the equation is worth more in the endgame and vice versa in the late opening or early middlegame.
I have seen expert players on several occasions win wonderful endgames with RPP v BB or RPP v BN. At the other extreme I have seen games in the middlegame stage where files are pretty much blocked but there one or two great diagonals and one player has sacrificed his rook for just the one bishop leaving him with a comfortable won game because his bishop is now a power piece and his opponent's rook is just a spectator. Similarly you will find many games online where one player sacrifices his rook for his opponent's knight on c3 or f3, doubling the pawns, smashing open the position in front of the castled king and leading to a quick win.
It totally depends on position. But if we say there is no tactical opportunity immediately, going into B+N vs R+p strategically would be about even.
Using Kaufman piece values, the value of B+N (assuming that you are also sacrificing the bishop pair and lose the pair bonus) is 2*3.25+0.5=7.0. The value of Rook and pawn is 5.0+1.0=6.0.
Thus, on purely material grounds, the exchange is bad for the side who gives in Bishop and Knight. Positional factors can make the exchange even worse or somewhat better.
