Why is this the solution?
The key is that killing the black stones doesn't give White an eye.  After White captures the two stones, Black will throw in at the same place he played in #3.  White can capture that stone too, but later in the game the stone above the capture will be placed in atari, and White will have to fill in the "eye" on the right.  Because that can happen, it is a false eye, and the White group is dead.