For triangulated categories the notion of Bousfield localization is a "special case" of the notion of Verdier quotient. As you observe (and is shown in Lemma 3.1 of the paper you mention) any Bousfield localization arises as the composition of a quotient functor and a right adjoint to the quotient functor. One can view the point of this as being that by general nonsense the right adjoint to a Verdier quotient is always fully faithful and so one can perform the localization by taking an appropriate subcategory of the category in which one started.  

Indeed suppose we have a fully faithful exact functor of triangulated categories R -> S where lets say (just to keep everything easy) S is compactly generated, so in particular has all small products and coproducts, and R is localizing then the following statements are equivalent (this works more generally):  
i) R -> S admits a right adjoint;  
ii) The quotient functor S -> S/R admits a right adjoint

and in this case the right adjoint S/R -> S identifies the Verdier quotient with the full subcategory R^{\perp} of R-local objects. We also have that the right adjoint S -> R identifies R with the Verdier quotient S/R^{\perp}.

By composing these adjoint pairs we get two functors S -> S one for each adjoint pair which are the acyclization and localization functors. These "project" objects of S onto their parts in R and S/R respectively (in the sense that our pair of adjoints define functorial triangles for objects of S presenting them as extensions of objects in S/R by objects of R and we project onto the corresponding object in this triangle) and this localization is the Bousfield localization with respect to the class of maps whose cone lies in R. Every Bousfield localization arises in this way which is the content of Lemma 3.1 in the paper of Benson-Iyengar-Krause.  
The conditions on the endofunctor L in Lemma 3.1 boil down to the fact that they are necessary for L to be idempotent and to arise as the monad associated to the unit of an adjunction.

If you wish I can provide references/more details later when I have a bit more time - essentially the point is that in a lot of situations we get Bousfield localizations from our quotient (for instance if R satisfies Brown representability and is localizing, or if it is localizing the quotient is essentially small and S satisfies Brown rep) and whether one uses the localization sequence (i.e. the pair of adjoints) or the localization and acyclization functors is a matter of which is most convenient and personal choice - the formalism tells you that they are equivalent.