Why is Black 1 in the solution diagram such an important point ? Well, the first thing to notice perhaps is that there is no eye in the corner:
a and b are miai to falsen the eye. We see there is no eye in the corner and that our killing move must be sought elsewhere.
Next, there is some eyeshape at the outer part of the group.
If White gets both marked stones in place, she makes miai of a and b to form an eye.
Let's take a closer look at the eyeshape at the inside.
The points marked a need two moves to become an eye. The points marked b need three moves to become an eye.
There for, one is inclined to call White a here the vital point. Let's try this defense against the proposed killing move.
Let's look at some other moves for Black.
In the last two variations, Black had to come back and destroy the eye at the outside. This explains already the virtues of the killing move. Let's see why the other moves destroying that eye don't work as a first move.
Now White can play at this vital point 2 and answer 3 at 4. The two marked stones have one liberty more than in the solution diagram, so Black 5 at a is not atari. White lives.
--Dieter