I have a question on definition/motivation of Virasoro algebra. Recall that Virasoro algebra is an infinite Lie algebra generated by elements $L_n$ $(n\in \mathbb{Z})$ and $c$ over $\mathbb{C}$ with relations $$ [L_m,L_n]=(m-n)L_{m+n}+\frac{c}{12}(m^3-m)\delta_{m+n,0}. $$ A typical explanation of this definition is the following.

Define vector fields $l_n=-z^n\frac{\partial}{\partial z}$ on $\mathbb{C}\setminus \{0\}$. They form a Lie algebra of infinitesimal conformal transformation $$ [l_m,l_n]=(m-n)l_{m+n}. $$ So the Virasoro algebra is a central extension of this algebra by $c$. $c$ is called the central charge.

My questions are

- How can one see that the Lie algebra above is associate to infinitesimal conformal transformation?
- What is the central charge $c$ intuitively? Why are we interested in such a central extension?

As to second question, I don't have enough physics background to check what the central charge $c$ means in physics literature.

At this point, I don't have any intuition and have trouble in digesting the concept. I would really appreciate your help.