What is the reason behind playing gambits on defence, such as in the French Defence?
The fact that an opening has the word "Defense" in the name usually signifies nothing more than the fact that it is a move by Black that leads into that opening. For instance, if White plays
1.c4 it is simply called the English Opening. But if, after
1.e4, Black plays
1...c5, then we reach Sicilian Defense positions; or if Black plays
1...e6 instead it is a French Defense. But the only reason the latter two are called "defenses" is because it is Black who chooses to enter each one, and the first-move advantage for White means that in the early stages of the game, Black is "defending" to some degree. This is the same reason why, for instance, when Black plays a certain setup it is called the King's Indian Defense, but when White does the same it is called the King's Indian Attack. The word "defense" in the name of an opening generally means no more than that Black has the choice to enter into it.
But Black's motives when playing one "defense" instead of another vary as widely as chess players do, and many times Black is striving to take the initiative away from White immediately. The Sicilian Defense and the King's Indian Defense, for example, both have rather aggressive reputations. Given the above explanation regarding the word "defense" in the names of openings, your question as stated boils down to, "what reasons does Black have for gambit play?" And to that, I would just say that gambit play has the same pluses and minuses for either color.
It's about imbalances. Some positions are very balanced, not many differences between both positions -- same material, castling on the same side, similar pawn structures. That's one popular strategy as black: keep the number of imbalances low, and slowly equalize the position. Only when it's equal do you try to outplay your opponent.
The other philosophy is to play for *un*balanced positions. Emphasize the differences. In the Open Sicilian, especially when White castles long, White gets quick easy development and an attack on the king, but Black has two pawns in the center and long term chances in the counter attack. In the King's Indian with a locked center, often White has an overwhelming attack on the queenside and Black a great attack on the kingside. It's less important who is currently slightly better, because the positions are like apples and oranges. What matters is who wins first.
Gambits are just another imbalance -- one side has an extra pawn, the other has some form of compensation. Maybe development, maybe something else.