As I understood from wiki page,
Given a finite group acting on a vector space, Molien series gives a generating function, although I am not sure what this means. And how is this related to the polyomial invariants of the group? Could anyone help me here?
The wikipedia page turned out to be quite useful, but also brief and a little faulty in some places, so I will rephrase this a little and correct wikipedia later.
Let $G$ be a finite group acting on a finite dimensional complex vector space $V\cong\mathbb{C}^n$ via $\rho:G\to\operatorname{GL}(V)$. We choose a basis $\vec{x}_1,\ldots,\vec{x}_n\in V$ and denote by $x_1,\ldots,x_n\in V^\ast$ the dual basis. Then, the polynomial functions on $V$ are quite literally the polynomial ring $\mathbb C[V]=\mathbb C[x_1,\ldots,x_n]$. The homogeneous polynomials of degree $d$ are $\mathbb C[x_1,\ldots,x_n]_d=\mathbb C[V]_d = \operatorname{Sym}^d V^\ast$, the $d$-th symmetric power of $V^\ast$. I am setting up this notation so that the wiki page is more comprehensible.
Now $G$ acts on $\mathbb C[V]$ via the action $G\times\mathbb C[V]\to\mathbb C[V]$ defined by $g.\phi:=\phi\circ\rho(g)^{-1}$. Homogeneous polynomials of degree $d$ are mapped to homogeneous polynomials of degree $d$ under this action, so $\mathbb C[V]_d$ is a finite dimensional $G$-representation. If we denote by $n_d := \dim_{\mathbb C}(\mathbb C[V]_d^G)$ the dimension of its invariant subspace, then the Molien series is the generating function of the sequence $d\mapsto n_d$. Note that a generating function of a sequence is a general concept, it is the power series $H(T)=\sum_{d=0}^\infty n_d T^d$ in the variable $T$.
Geometrically, $\mathbb C[V]_d^G=\mathbb C[V/G]_d$ is the $d$-th graded part of the coordinate ring of the geometric quotient $V/G$, so $H(T)$ is the Hilbert series of the variety $V/G$.
Did this explain how everything is related? Feel free to ask.