Your questions are a bit vague so if my answer isn't satisfying please specify what exactly you want to know. I am also still struggling to get my head around the concept and I am far from an expert but here is what I know.
So the Chern-Simons term, \begin{equation} -\theta_{ab} F^a_{\mu \nu} \tilde{F}^{a \mu \nu}. \end{equation} is present because it can be. This might seem ad-hoc but we are trying to construct the most general gauge-invariant Lagrangian. Having said this, there is also a method of introducing such a term through anomalies. The term is actually equal to a divergence, the Chern-Simons current, \begin{equation} F^{a}{ \mu \nu} \tilde{F}^{a \mu \nu} = \partial\mu K^\mu, \quad K^\mu = 2 \epsilon^{\mu \nu \rho \sigma} \left( A_\nu^a \partial_\rho A_\sigma^a + \frac{g}{3} f^{abc} A^a_\nu A^b_\rho A^c_\sigma \right). \end{equation} This implies that the term has no influence on perturbation theory, but can create non-perturbative effects. This is why there is no such term in QED.
One way of interpreting this term and where it becomes important is in the study of the ground state. The ground state of a gauge theory is pure gauge, i.e. \begin{equation} A^\mu = U^{-1} \partial U. \end{equation} This is obvious because $A^\mu = 0$ is definitely a ground state and we can still perform gauge transformations. 298
Now consider $U \in SU(2)$ and work in the temporal gauge $A_0 = 0$. Then we can identify $SU(2)$ with the 3-sphere $\mathcal{S}3$ and we can also identify $A^i \in \mathbb{R}^3$ with the 3-sphere for example through a stereographic projection. Thus a gauge transformation is a mapping $\mathcal{S}_3 \longrightarrow \mathcal{S}_3$. These kinds of mappings are classified into homotopy classes distinguished by the winding number $\nu$. One can show that the difference between two winding numbers is \begin{equation} \nu_2 - \nu_1 = - \frac{g^2}{32 \pi^2} \int{t_1 \le x^0 \le t_2} d^4x\; F^{a}{\mu \nu} \tilde{F}^{a \mu \nu}. \end{equation} So here the Chern-Simons current becomes important again. Transitions between different winding numbers must therefore proceed through non-vanishing $F^{a\mu \nu}$. The vacuum states can then tunnel through these potentials. The true vacuum of a theory is then a superposition of states with different winding numbers. If one takes this into account, then the Chern-Simons term has to be included naturally in the Lagrangian, \begin{equation} \mathcal{L}\text{eff} = \mathcal{L} - \theta \frac{g^2}{32 \pi^2} F^{a \mu \nu} \tilde{F}^a_{\mu \nu}. \end{equation} Furthermore, the energy of the vacuum state is dependent on the mixing angle $\theta$.
By the way, this does not occur classically because it relies on the tunneling effect which is only a feature of quantum mechanics. 298