What is variation of the Chern-Simons functional, and why can it be calculated as follows?

Let $$G$$ be a Lie group. Assume that we have an Ad-invariant bilinear symmetric form $$\langle-,-\rangle : \mathfrak{g} \times \mathfrak{g} \to \mathbb{C}.$$ Given a smooth manifold $$X,$$ we let $$\mathcal{M}_X$$ denote the moduli space of flat connections on principal $$G$$-bundles, modulo gauge equivalence. One can then show that the Chern-Simons action determines, for each closed surface $$Y,$$ a hermitian line bundle $$\mathcal{L}_Y \rightarrow \mathcal{M}_Y$$ with a connection $$\theta.$$ Further, for each compact oriented $$3$$-manifold $$X,$$ we get a parallel section $$s:\mathcal{M}_X \rightarrow r^*_X \mathcal{L}_{\partial X}$$ of the pullback bundle by the restriction $$\mathcal{M}_X \to \mathcal{M}_{\partial X}.$$

I have read that the fact that $$s: \mathcal{M}_X \to r_X^* \mathcal{L}_{\partial X}$$is a parallel section implies that the variation of the Chern-Simons action along a path of connections can be computed through parallel transport. This should be done by integrating the connection $$\theta$$ along the restriction of the path to the boundary. My questions are:

1. What does "variation of the Chern-Simons action" along a path of connections mean in this context?
2. Why can it be computed as claimed?