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 .

**Update**:

Let me try another explanation. We know from the above that
$$
n S_{CS} = \int_B F \wedge F \quad \mbox{mod 1}
$$

Thus,
$$
S_{CS} = \frac{1}{n}\int_B F \wedge F + \frac{q(B,E)}{n}
$$
with $q(B,E) \in \mathbb{Z}$. The simplest guess is that $q = 0$, but it's easy to see that the resulting action is not independent of the choice of $B$. In particular, we would want, for closed $B$, that $\frac{1}{n} \int_B F \wedge F \in \mathbb{Z}$, but it's only in $\frac{1}{n}\mathbb{Z}$.

So, the goal is to choose a $q(B,E)$ such that the action makes sense. Since you want something that is an integer when applied to a closed $B$, it's not too hard to guess something like DV's action.