The first idea we explore involves putting a heavy piece (a queen or rook) on h8, where it mates with diagonal protection from the bishop or perhaps from a pawn on g7. The diagram to the left shows in skeletal form the position you are trying to reach. The rook on h8 could as easily be a queen, but when it is a rook the division of labor between the attacking pieces is particularly elegant: the rook covers the light squares the bishop can’t reach; the bishop covers the dark squares and protects the rook. Black is mated.
There are three general elements needed to create this pattern: (a) An enemy king more or less stuck in the corner because f7 and perhaps f8 are blocked off as flight squares, usually by the king’s own pawn and rook as in the diagram. This is a common state of affairs soon after a player has castled. (b) An open line to h8 for your bishop because Black’s g-pawn has moved forward. (c) An open h-file so that one of your heavy pieces can be dropped to h8. In practice you often need to make sacrifices to create these open lines; as a result this pattern and those that follow frequently require that you start with two of your pieces available to attack on the h-file—maybe two of them lined up on the file from the start of the attack, or one piece on the h-file and another that can get there in one move. The first piece is sacrificed to clear the way for the second, which is ready to go or arrives shortly, preferably with check.
“Anderssen’s mate” generally refers to one particular type of sacrifice used to open up the h-file for this purpose; it is named after a famous game of Anderssen's we will see in a moment. But you also might as well associate Anderssen with this general point to make it easier to remember: if you have a diagonal attack against h8, you should ask whether you might mate there with a heavy piece—even if it takes large sacrifices to get it done.