Green functions in QFT What is the sense of Green function
$$
\langle | \hat {T}(u_{1}(x_{1})...u_{n}(x_{n})\hat {S})|\rangle , \quad \hat {S} = \hat{T}e^{i\int \hat {L}(x)d^{4}x} ?
$$
How is it connected with scattering processes?
 A: The Green's function you have given arises as one step in the computation of elements of the S-matrix in the LSZ-formalism, if one considers the S-matrix in the interaction (or Dirac) picture.
Our goal is to compute scattering amplitudes of the form $\langle \prod_i p_i, \mathrm{in} | \prod_j q_j,\mathrm{out}\rangle$. In the course of the computation, we reduce this to the n-point function of the full Heisenberg fields $\langle \Omega | \hat T\prod_{k} \phi(x_k)| \Omega\rangle$. Switching to the interaction picture, we relate the Heisenberg fields to the interaction field through the time evolution operator $\hat U(t_1,t_2)$ fulfilling the relation $\mathrm{i}\partial_t \hat U(t_1,t_2) = \hat H_I(t_1)\hat U(t_1,t_2)$ which is solved by the Dyson series 
$$ U(t,t_0) = \sum_n (-\mathrm{i})^{n} \int_{t_0}^t \mathrm{d}t_1 \int_{t_0}^{t_1} \mathrm{d}t_2 \dots \int_{t_0}^{t_{n-1}} \mathrm{d}t_n \hat H_I(t_1)\dots\hat H_I(t_n) = \\\sum_n \frac{(-\mathrm{i})^{n}}{n!} \int_{t_0}^t \mathrm{d}t_1 \int_{t_0}^t \mathrm{d}t_2 \dots \int_{t_0}^t \mathrm{d}t_n \hat T(\hat H_I(t_1)\dots\hat H_I(t_n)) = \hat T \mathrm{e}^{-\mathrm{i}\int_{t_0}^t H_I(t')\mathrm{d}t'}$$
This is essentially your $\hat S$, only that instead of the Lagrangian density we have the Hamiltonian density (I don't know a case where the Lagrangian appears from the top of my head).
In the interaction picture, one finds (after a tedious calculation) that
$$ \langle \Omega | \prod_{k} \phi(x_k)| \Omega\rangle = \lim_{T\rightarrow\infty}\frac{\langle 0 |\hat T \prod_k \phi_I(x_k)\hat U(-T,T)|0\rangle}{\langle 0 | \hat U(-T,T)|0\rangle}$$
This way, the function you have given is (up to normalization) essentially an n-point function, which in turn is essentially a scattering amplitude.
The entire computation and reasoning can be found in any decent QFT book, I have only tried to motivate the appearence of this term. Please let me know if you want more detail on something (or if this wasn't helpful at all).
EDIT: You are right, the $|p,\mathrm{out}\rangle$ states are fock states in the asymptotic Hilbert spaces. But, inverting the mode expression for the free field, one finds that
$$a_{in}(\vec{p}) = \frac{\mathrm{i}}{\sqrt{2E_q}}\int \mathrm{d}^3x \mathrm{e}^{iqx}\overset{\leftrightarrow}{\partial_0}\phi_{in}(x)$$
where $x^0$ is arbitrary (this is another tedious calculation) and $f \overset{\leftrightarrow}{\partial_0} g := f(\partial_0 g) - (\partial_0 f)g $. By this, you can write 
$$\langle \Omega,\mathrm{in} | p,\mathrm{out}\rangle = \sqrt{2E_p}\langle \Omega,\mathrm{in} | a_in(p) | \Omega,\mathrm{out}\rangle = -\mathrm{i}\int\mathrm{d}^3 x\mathrm{e}^{ipx}\overset{\leftrightarrow}{\partial_0}\langle \Omega |\phi_{in}(x)|\Omega \rangle$$
. By repeatedly applying this, you can reduce the question of $\langle \prod_i p_i, \mathrm{in} | \prod_j q_j,\mathrm{out}\rangle$ to the n-point functions.
