It seems easier to motivate the solution by first showing this reference diagram, which is actually a failure for White.
White would like to pull out the White stone with . But and can capture the three White stones in a connect-and-die, creating two eyes in the process. The question is then, how can White prevent this connect-and-die? The trick here is to reduce Black's liberties.
More elaboration on the reference diagram. It should be noted that after , although Black cannot block at directly, Black can first capture the three stones at in sente, before return to block at . If is played at , the same comment applies.
Due to shortage of liberties, Black cannot connect and atari at a after , so the Black corner dies.
A main difficulty of this problem is to spot the first move in the solution. Playing here instead would be a mistake. Now can capture and threaten to make the second eye.