What is the main use of Lie brackets in the Lie algebra of a Lie group? I am beginner in Lie group theory, and I can't find the answer a question I am asking myself : I know that the Lie algebra $\mathfrak g$ of a Lie group $G$ is more or less the tangent vector of $G$ at the identity, so that $\mathfrak g$ have a very interesting property : linearity.
However $\mathfrak g$ has another property : it is stable under Lie brackets $[.,.]$.
For me when I study Lie groups I always find linearity of Lie algebras really important, and I don't see and I didn't find why the stability under Lie brackets is important. What is the main result/property of Lie groups using this property?
That would be great if you could light me!

As a take-home message, I would say that the Lie bracket encodes the structure of group multiplication. The linear structure alone is very uninformative; it really just tells you about the dimension of the manifold.
More precisely, the Baker-Campbell-Hausdorff formula tells you that the group multiplication (at least for elements near the identity, and hence in the image of the exponential map $\exp$) can be entirely obtained from Lie brackets. Therefore, as pointed out in Sasha Patotski's post, (almost) all group properties have Lie bracket analogies.
In fact, the Lie bracket is particularly nice, giving very concise expressions of Lie group properties. Thinking about the adjoint representation, Killing form and so on is a good way in.
Remark: The caveats above reflect the problems like $\exp$ mapping the Lie algebra of $O(3)$ back to just $SO(3)$, and a more subtle problem where the Lie algebra of $SL_2(\mathbb R)$ is mapped to a strict subset of that group (even though the group is connected).
