White has a lot of compensation for the sacrificed exchange, specifically:
- more space: the white central pawns are perfectly placed to limit the movement of the black knight and bishop
- play on the dark squares: black has considerably weakened the dark squares on the king side and does not have a dark-squared bishop anymore; White can easily take advantage of that by operating on these squares with bishop, knight and queen.
- lead in development: Basically white only needs to activate the rook, while black really struggles to free himself. Also black will be forced to move the rook on h8 (castling seems too dangerous), losing another move for development and leaving the king in the center for some more time where it will be under attack.
This suggests that white can make use of these elements, before black can recover.
Somewhat concrete lines:
Black's rook h8 is under attack and has to move. Three options:
- 1...0-0 looks very dangerous for black. White could just play something like Qd2, Qh6, Ng5 and if necessary h4, h5. I don't see what black could do to avoid getting mated soon. Note that black cannot play something like f6 soon because of the discovered check from the b3 bishop.
- 1..Rg8 or 1..Rf8. Here white has basically two ideas, which can be combined in various ways. a) play d6 which prevents black from playing d6 himself (freeing the bishop on c8) and allows white to activate the queen (via d6-f6). Also, with the white queen on d6 and the white bishop on f6, white would basically bind four black pieces (Qd8, Ne7, Ke8, Rf8) to very passive positions. b) play Ng5, which hints at an attack on f7 and also on re-routing the knight via h7 to f6.
So overall, yes the position is pretty hopeless for black.
Regarding your general quesion (Why are these things worth three pawn points? And how do I know when they are worth that much?):
You should realize that material is only one of many factors relevant for judging a position. Admittedly, in many games, players have somewhat symmetric positions, are developed equally well, etc and in this case, material becomes a major factor. So yes, in quiet normal positions material is often very important.
However in highly unbalanced positions like the one in your example, material count becomes less important and is easily trumped by activity. There is no strict rule to know when this is the case. In fact this is among the more difficult aspects of chess and can basically only gained by experience. I would strongly discourage you from trying to assess such unbalanced positions by summing points for "piece activity", "weak king"... Chess just does not work this way.
Computers do something like this in their evaluation function, but they do this for many (lots) positions many moves down the line (not only the current position) and then it does make sense. Furthermore computers have a more fine-grained number for "space", "piece activity", etc.
Anyway, humans lack the power to do all this and basically need to rely on experience, intuition and (occasionally) calculating some more or less forced concrete lines.
As a very rough guideline (don't follow this strictly) you can sacrifice:
- a pawn: to gain a lead in development/piece activity or central control, for at least a short amount of time, often with a potential to regain the pawn easily if needed (see e.g. Queen's gambit). Sacrificing a pawn is not a major deal as many positions are still a draw +- a pawn.
- up to three pawns or an exchange: if you get a major long term positional advantage through it; often you end up with a very strong minor piece vs. a not-so strong rook (e.g. in somewhat closed positions). Here you really need to have something concrete.
- more than three pawns or an exchange: rarely, typically in direct mating attacks only