As I learn more about the correspondence between Lie groups and Lie algebras, the underlying structure of a Lie algebra feels more and more mysterious to me. It seems that the Lie bracket is absolutely necessary to do anything with the Lie algebra, but for a while I had no idea what it was even doing.
Recently, however, I found an interpretation which seems to get close. Let $G$ be a Lie group and $\frak{g}$ the underlying Lie algebra, treated as the tangent space at the identity $e$. Define the conjugation map $\gamma_g: G \to G$ for each $g$, and define $\text{Ad}: G \to \text{Aut}(\mathfrak{g})$ so that $\text{Ad}(g) = (\gamma_g)_{*e}$. Further, differentiate this map at $e$ to get $\text{Ad}_{*e} = \text{ad}: \mathfrak{g} \to \text{End}(\mathfrak{g})$. We define $[X, Y] = \text{ad}_X(Y)$. According to this definition, the Lie bracket is the infinitesimal version of conjugation; roughly speaking, it tells you how much the infinitesimal $Y$ changes if you conjugate by the infinitesimal $X$.
This gets so close to being a satisfying explanation! There's just one, very important, loose end: how is mere conjugation able to completely classify the (local) structure of Lie groups? It seems completely unreasonable to me that such a specific operation should be able to generate every element of the group. I've tried analysing the proofs of theorems in more detail, for example the classification of simply connected Lie algebras and the BCH-formula, but they all seem to either lose me in screeds of arbitrary choices or use results which aren't related to the adjoint representation.
My question is therefore the following: what is the significance of conjugation in the study of Lie algebras? How, intuitively, can we obtain the structure of a Lie group just from the way that conjugation acts near the identity?