They are trying to define the Chern-Simons action over a manifold $M$ by writing it as the integral of $\int F \wedge F$ over a bounding manifold $B$. When the bundle is nontrivial, they consider a more general cochain and show that there exists a $B$ over which the bundle extends such that $\partial B$ = $n$ copies of $M$. So, you can define

$$
n S = \int_B F \wedge F
$$

But, because actions enter into imaginary exponentials in the path integral, this is really only defined mod 1 (once you reenter all the coefficients that I omitted). So, the action $S$ is only defined mod $1/n$.

They show how the second formula resolves the ambiguity in the text that follows, but it's probably best to think of it as a differential character or in terms of differential cohomology. A more rigorous presentation might be http://arxiv.org/abs/hep-th/9111004.pdf .