quantum field theory - Intuition for S-duality

first of all, I need to confess my ignorance with respect to any physics since I'm a mathematician. I'm interested in the physical intuition of the Langlands program, therefore I need to understand what physicists think about homological mirror symmetry which comes from S-duality. This question is related to my previous one Intuition for Homological Mirror Symmetry

S-duality

As I have heard everything starts with an $S$-duality between two $N= 4$ super-symmetric Yang-Mills gauge theories of dimension $4$, $(G, \tau)$ and $(^{L}G, \frac{-1}{n_{\mathfrak{g}}\tau})$, where $\tau = \frac{\theta}{2\pi} + \frac{4\pi i}{g^2}$, $G$ is a compact connected simple Lie group and $n_{\mathfrak{g}}$ is the lacing number (the maximal number of edges connecting two vertices in the Dynkin diagram) . And, then the theory would be non-perturbative, since it would be defined "for all" $\tau$, because amplitudes are computed with an expansion in power series in $\tau$

So I need to understand what this would mean to a physicist.

1) First of all, what's the motivation form the Yang-Mills action and how should I understand the coupling constants $\theta$ and $g$?

2) How can I get this so called expansion in power series with variable $\tau$ of the probability amplitude?

3) What was the motivation to start looking at this duality? A creation of an everywhere defined (in $\tau$) gauge theory, maybe?

Thanks in advance.