In Chess, a player is said to be in check if the opponent would be able to capture this player's king. Players cannot make a move that puts or leaves their own king in check. This rule is called check-rule.
The white king can move only to the three fields marked by blue dots, because on the others, it would be in check from the rook or the king. Note that the two kings can never be on adjacent fields, because then they would be mutually in check from each other.
The bishop cannot move, because then it would expose the king into check.
When speaking about "the opponent would be able to capture the king" in check-rule, it is irrelevant whether the king-capturing move itself would be legal by the check-rule. For example:
The black king cannot move into check, i.e. it cannot move to the bishop's diagonal – not even if the bishop is not allowed to move by the check-rule, like in this case.
The white king cannot move next to the other king, because there it would be in check from that, i.e. the black king would be able to capture the white king. It is irrelevant that the black king can neither move there because of the white rook.
White is in check from the knight. Therefore, the bishop cannot move. Hence it cannot give check to the black king. It does not matter that the "knight captures the king" move would then be illegal, since the black king would be in check from the bishop.
There are three possible ways to escape from check:
1st way: moving the king.
2nd way: intervening an other piece.
3rd way: capturing the checking piece by an other piece.
Further concepts relevant to the check-rule:
Checkmate: the king is in check, and there is no legal move to escape from check. In this case, the game finishes, and the player who gave checkmate (in this case Black) wins.
Pin: a piece (in this case the bishop) cannot move, because it would expose its own king to check.
Discovered check: a piece (in this case the bishop) moves away, and discovers a check from an other piece of the same colour.
Double check: the king is in check from two opposing pieces at once. One of the two checks is neccessarily a discovered check. There is only one way to escape from a double check: to move the king.
Perpetual check: the white queen gives check repeatedly (from the two marked fields). It is useful when White cannot give checkmate, and is about to lose the game. Giving checks blocks the opponent's checkmate threat, and if checks can be given forever, the opponent can never give checkmate, so the game results in a draw.