What is the main use of Lie brackets in the Lie algebra of a Lie group?

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!