The and groups are in a capturing race. Both groups have four liberties and it is black's turn, so black should be able to win the capturing race.
However, if black begins filling outside liberties as normal, he runs into shortage of liberties. If black plays at , treating it as a necessary approach move, both groups still have four liberties and it is now white's turn, so black loses.
If black plays this and allows white to capture , white must spend another move to capture , and the result is not even ko, since black cannot play x due to shortage of liberties.
is tesuji-ish, threatening to connect out while reducing a liberty. does not gain any liberties, so it is okay that reduces liberties for both groups. This is interesting because normally shared liberties should be filled last. Now when white reduces liberties at (or white a) is does not require an approach move at b because it is atari on white's group.
If white plays this , attempting to connect out, black simply connects himself with . When white plays or at a, is atari and white doesn't have time to connect.
It may seem that and can be switched in the first solution diagram, but only works because of (from the solution diagram). If black reduces a shared liberty first, white simply reduces black's outside liberties and wins.
Bill: Oh, ye of little faith! saves the day.