Why is the f6 pawn break in Advanced French so important for black? I am new to the French (so forgive me if the question seems elementary).
I've just come across the pawn break ...f6 for black in the French advance variation, and I find the pawn break quite dubious for me for several reasons:


When we are attacking the opponent's pawn chain, we are told that to attack the pawn chain, the best way is to attack its base. But in this case, black seems to be voluntarily giving up his pawn chain, as the base of the f7-e6-d5 pawn chain is at f7. Isn't black self-destructing his pawn chain in advancing the f-pawn, as white can now attack the e6 pawn?
After the inevitable exchange of the pawns (exf6 Rxf6), to me, black is left with a backward pawn (e6) and it is easily exploited by white. My question is: how is the backward pawn favouring black's position, and why is gxf6 not played, as black can get rid of the backward pawn by playing e5 on the next move.
If it's argued that f6 develops the black light-squared-bishop by the manoeuvre Bc8-d7-e8-h5, why can't black just play b6 and fianchetto the bishop or even play Ba6?


Thanks in advance.
I am new to french (so forgive me if the question seems elementary) I've just came across the pawn break f6 for black in the french advance variation, and I find the pawn break quite dubious for me for several reasons


You must understand one whole new concept of chess modern strategy : Weakness is not a weakness if it can not be exploited. 

And another thing: Opening theory chooses only the strongest moves as mainlines. 

These moves are the strongest moves known today, so if Black is willing to play with a backward pawn, then be sure he has very good compensation for that!

So you have a backward e-pawn, so what? What are the other features of the position? 


Can you get active piece play for this weakness?
How easy is for White to attack it? Does he have to give up "something" ( development/control of a key square/etc ) in order to attack it?
Can you sacrifice that pawn in the near future for strong pressure/initiative?
Is it really a weakness or is it simply ugly?


Once you change your point of view like this, you will see the other potentials that you have totally neglected or simply underestimated. 


  When we are attacking the opponent's pawn chain, we are told that to attack the pawn chain, the best way is to attack its base. But in this case, black seems to be voluntarily giving up his pawn chain, as the base of the f7-e6-d4 pawn chain is at f7. Isn't black self-destructing his pawn chain in advancing the f-pawn, as white can now attack e6 pawn?


Can White really attack the e6 is the question here that is important. The modern practice proves that White can not do this advantageously. I have provided an explanation why, and have posted some examples below.


  After the inevitable exchange of the pawns (exf6 Rxf6), to me, black is left behind with a backward pawn (e6) and its is easy to be exploited by white. My question is: how is the backward pawn favouring black's position, and why is gxf6 not played as black can get rid of the backward pawn by playing e5 on the next move.


Black has clear math behind this:


White has lost a tempo with e5 by playing twice with a pawn;
White is passive and for now only maintains space advantage;
Because of #1 and #2, White needs time to finish development so he can "cash in" his space advantage;
White pieces are slightly misplaced -> Nb1 belongs at c3 but the pawn is there, Bc1 should go to g5 or e3 but is hindered by ...Qb6;


Now the important part:

If Black removes d4 + e5 chain then his superior development will be felt and he will get mobile central phalanx -> mobile center phalanx + superior development/piece placement = high chances to win. 

This is very stressful moment for White, and he needs accurate play just to survive. He needs to finish his development but his strong pawn chain is gone. That is why he has no time to attack e5, it is just too dangerous. 

So you see, Black will follow ...f6 with ...c5 in order to get his central pawns rolling, and since he will get his pieces ideally placed, the initiative he obtains might be decisive. 

Furthermore, f-file + Qb6 + Nf6 + Bc5 make tremendous pressure at f2. There is thematic ...Nxf2/Bxf2 sacrifice with subsequent ...e4 that regains the piece but retains the initiative. You have never mentioned that you are aware of this plan! Without knowing these plans it is easy to look at Black strategy as bad. Now you understand why ...gxf6 is usually a mistake.


  If it's argued that f6 develops the black's light-squared-bishop by the manoeuvre Bc8-d7-e8-h5, why can't black jus play b6 and fianchetto the bishop or even play Ba6?


On b6 usually sits Black queen, so the plan you mentioned fails. She is so well placed there, while ...Ba6 would achieve little. With the pawn center rolling ( after Black plays ...c5-cxd4 and ...f6-fxe5 ) bishop will come into play just in time, and on a good g4 or f5 square with strong effect.

PRACTICAL EXAMPLES OF ABOVE STATEMENTS:

I will quote some theoretical lines form the Advance French just to show you above plans in action. The sample lines are quoted from J.Watson-Play the French 4th edition ( 2012 ) :

[Title "Black demolishes White pawn chain and obtains dynamic piece play"]
[fen ""]

1.e4 e6 2.d4 d5 3.e5 c5 4.dxc5 Nc6 5.Nf3 Bxc5 6.Bd3 f6 7.Qe2! fxe5 8.Nxe5 Nxe5 9.Qxe5 Nf6 10.O-O O-O ( 10...Bd6 11.Qg5 O-O 12.Nc3 e5 13.f3 Be6 ) 11.c4 Ng4! ( 11...Qb6!? ) 12.Qh5 Rxf2 13.Rxf2 Bxf2+ 14.Kh1 Nf6 15.Qe2 Bb6


"and White has some, but not fully sufficient, compensation for the pawn."

Also notice that after 11...Qb6!? sideline if White plays 12.Qe2 Black plays 12...e5! threatening both ...e4 and ...Ng4. 

Here is another example:

[Title "Black demolishes White pawn center and obtains equal position"]
[fen ""]

1.e4 e6 2.d4 d5 3.e5 c5 4.c3 Nc6 5.Nf3 Qb6 6.Be2 f6 7.O-O fxe5 8. Nxe5 Nf6 9.dxc5 Bxc5 10.Nd3 ( 10.b4? Bxf2+! ) 10...Bd6 11.Be3 Qc7 12.f4 O-O 13.Nd2 e5 ( 13...Bd7! ) 14.fxe5 Nxe5 15.Nxe5 Bxe5 16.Nf3 Bd6 17.Kh1 Bg4 18.Bg5 Kh8 19.Bxf6 Rxf6 20.Nd4 Bxe2 21.Qxe2 Bxh2 22.Rxf6 gxf6 23.Qe6=


The whole point of ...f6 is to demolish White pawn chain so you can start rolling your own central pawns and exploit your development advantage/better piece placement. After ...f6 make sure you can follow up with ...c5 and just open the game. You should be just fine after that.

This is the plan that was not familiar to you, but this plan is the one that "holds" this line "together". If you need further help leave a comment. Best regards.