The last displayed formula on page 41 of the lecture notes does not defined a section of the complex line bundle $L$. It is rather a definition of $L$. It states that if $C$ is a point of $M$, i.e. a flat connection on $\Sigma$, then for every 3-manifold $B$ such that $\Sigma = \partial B$ and for every $A$ connection on $B$ whose restriction to the boundary is $C$, the quantity $e^{2i \pi S(A)}$ is an element of the fiber of $L$ at $C$ (where $S$ denotes the classical Chern-Simons functional). This statement does not define a section because it does not say what to associate to a point $C$ of the basis but what do associate to a point $C$ and to a connection $A$ of boundary $C$.

The way to define a section is quantum and is recalled in the question. To define a section from the procedure of the preceding paragraph, one has to eliminate the dependence on the choice of $A$. But quantum field theory does that for us because it tell us to integrate over the all possible $A$. In other words, if $B$ is fixed, a section $v$ of $L$ is defined by assigning to a point $C$ the quantity defined formally by the path integral (over the space of all gauge equivalence classes of connections $A$ on $B$ of boundary $C$):

\[ \int_{A, \partial A =C} DA e^{2i \pi S(A)}, \]

which is indeed an element of the fiber of $L$ at $C$ because it is a "sum" of such elements by the definition of $L$.