king move

A King can move one square in any direction. It can not move into a square that an enemy piece attacks (it can not place itself in check).

Of course it can!

You wrote in the original comment 'diagonal to pawn' without reference to the king being diagonally in front of the pawn.

Of course the king cannot move to a square in which the pawn attacks (i.e. diagonally in front).

The objective of the game is to capture the opponent's king. However, practically speaking, we do not actually capture the kings.

Moving the king to an attacked square (such as a pawn's capturing squares) is forbidden by the rules of chess because the king would be captured by the pawn. So in practical situations, we are not allowed to move the king to an attacked square, and we do not capture kings at all.

He’s giving an example of it. 

What “dangerous zone” is this? 

The game always ends by checkmate if the King is in a position where he cannot escape capture.

So no, the King cannot be captured... the game ends before that can happen.

The black pawns are moving down the board in the second game, and that black pawn captures on d3 and f3

Right! But not "in-front-diagonal".

Well, you could move in "front-diagonal' to your own pawn.

but I still dunno...

What's the confusion? The King can not move into any square that an enemy piece attacks (it can not place itself in check). Simples. You can not put the king on ANY square where the opponent could capture it on their next move.

You're right

I KNOW that.

Then what's the confusion? 

<(MEH.)

You did not answer my question.