The f6 break is a key idea in the French defense, and the point is mainly to add more pressure to White's center. In the French, one basic line of thought is the following:
- Let White occupy the center
- Destroy White's center
- Occupy the center yourself
- Win the game, as White now will be totally helpless!
If black manages to complete steps 1-3 outlined above, without significant material losses or a serious weakening of the king's position, step 4 will indeed often be possible to accomplish.
Let us examine the move f6 in view of the steps above. If White has a pawn on e5 and black plays ...f6, it's possible to
- a) take on f6
- b) defend the e5 pawn
- c) ignore it
a) If Black hasn't castled, it most likely means that Black will have to respond to this with ...gxf6. Black now typically has pawns on e6 and on f6. It is not hard to see that Black can have plans to push ...e5 in the near future, and if the pawn capture on f6 hasn't left Black's king open to direct attack, it's possible that Black will achieve steps 2-3 outlined above shortly.
If Black has castled, on the other hand, more options than ...gxf6 will be available. Often either a knight, bishop or a rook could also take on f6 in this case. In case of rook capture, Black has a semi-open f-file for his/her rook. Taking with a bishop adds more pressure on the a1-h8 diagonal. Taking with a knight adds possibilities to play ...e5 to free the black bishop on c8 in some cases. It is clear that the ...f6 break by Black has given many possibilities for future plans.
One more note: If White takes the pawn on f6, the h2-b8 diagonal will be ripe for the black bishop to occupy, often coupled with a queen on c7. This is sometimes very unpleasant for White.
b) If White protects the pawn, Black has some other options with ...f6 played. If White has played f4, Black could sometimes take on e5, and if White recaptures with the f-pawn, it can in some cases be possible to play ...Qh4+ for some concrete tactical purpose. Other times Black can be perfectly happy with just trying to take over the f-file and pressure White's position that way. If White recaptures with the d4 pawn instead of the f-pawn, Black has the option to open the diagonal a7-g1 with ...c4, or play ...d4 and just try to break once again in the center, to activate his/her pieces.
If White has played f4, and it's not good for Black to take on e5, Black can wait, and try to maneouver the c8 bishop to g6 via Bc8-d7-e8-g6. This is a standard plan in the French defense to activate that light squared bishop. Or Black can lock the center with ...f5 in some cases, to stop White from attacking the black king. In rare cases, after White has played f4, Black can even try to play ...g5, to again just try to break up White's center. This can be very risky, so it's not often the first idea Black thinks of, but it should never be forgotten, as it can recieve a great pay-off if well executed.
If White guards the e5 pawn with Bf4, the g5 break is more natural, and Black can even try to go for a kingside pawn storm in some cases. Again, it has to be well executed, as it can easily expose Black's king. Otherwise it's more typical for black to keep the tension in the center, and let White make the big decision to alter the status quo. However, at every move, White has to be wary of possible alterations of the center. The same is usually true if White defends e5 with another piece.
c) This option is very rare, and can only be considered if White manages to create an enormous treat elsewhere on the board that black needs to deal with immediately. If this is the case, the ...f6 break may be a bad idea. Otherwise, if White ignores the ...f6 break, Black can usually just take on e5, and normally having a knight on c6 and a queen on c7 (and maybe even a knight on d7 in some cases), this can mean that black simply wins a pawn.
This reply got much longer than I originally had intended it to be, but I feel like most of the reasons why Black may consider playing ...f6 in the French defense are in here. Remember that I have not included any concrete examples here, and that it all depends on the position. If ...f6 is an obvious blunder, for some reason I failed to mention here, it should of course not be played.
TL;DR: ...f6 is about challenging White's control over the center, and trying to open up the position for Black's pieces.