Need Help

In the above position black played Ka5.....author states that this played to win control of b4 square......

I wanna know why controlling b4 square is so important for black????

After Kb4 black can play the pawn to a4. After the pawn exchange the black king is on the 4th rank. The 4th rank is important because white's weak d pawn is on the 4th rank. In such situations where the pawn and king are on the same rank, the pawn is inevitably lost... there is no way to defend it (you can test this for yourself).

Two common patterns here to learn. First is being on the same rank as a weak pawn in an endgame guarantees you can win the pawn. Second is the idea with Kb4 and then bringing the pawn up to a4. By exchanging pawns in this way the king gains access to the squares the enemy pawn had been protecting.

Both of these ideas are seen in many pawn endgames.

In addition to what was already said in post 2, you have to learn the concept of Opposition.

Black occupies b4 and advances the a-pawn while the White King toggles.

Black trades and his King is now on a4.  With control of c4 via the d5 pawn, White can not stop black from eventually winning the d-pawn.  Let's say White moves Kc3 after Black takes on a4.  Then Black moves Ka3, White must move his King away from b4, giving Black Kb4 the next move.  Then White goes Kd3 to keep Black out of c4.  No good, Black then plays Kb3, White must move the King, and Black goes Kc4.  Then after Ke3, Black plays Kc3, and White must move away from the d-pawn, and the d-pawn falls, and Black wins.

Note that if White ever gained the opposition, Black had the extra move, g7-g6.  It is critical not to waste these moves as you may need to regain opposition at some point.  White has no such spare move, and so he could never gain the opposition from Black.

A very important concept that many never get is don't waste moves on pawns that don't need to move yet.  Keep it for when it's necessary to lose a tempo.