There could be different ways to give meaning to the phrase "explicit construction".
In an algebro-geometric sense, an expicit construction comes from more classical Borel-Weil-Bott theorem of which BDD is an abstract generalization. There's a number of proof in the literature, e.g the one by Jacob Lurie (on his home page).
According to the BWB, you can get the (finite-dimensional) representation by taking the global sections of one of the equivariant bundles $ \mathcal O(\lambda)$.
Another way to construct the representation would be to start with some simple $\mathfrak g$-modules and combine them to get your represenation. In this way, BDD helps by establishing a correspondence between simple equivariant D-modules and Verma modules. Therefore, the resolution for a bundle $\mathcal O(\lambda)$ corresponds to a construction in the category of $\mathfrak g$-modules, the one called Bernstein-Gelfand-Gelfand resolution, giving rise to Weyl character formula
As an example, the $\mathfrak{sl}_2$ modules correspond to equivariant D-modules on a $\mathbb P^1$, which has two cells. Therefore, a BGG resolution for an $\mathfrak{sl}_2$-module has two terms. Since a Verma module for $\mathfrak{sl}_2$ with an integer weight $\lambda$ has (I think) exactly one vector of each weight $\lambda' \le \lambda$, you can picture it as a ray on a weight lattice; the picture then becomes [segment] = [ray] - [ray].