This rather literary and semi-archaic word for woody undergrowth or a mass of trees or shrubs ultimately comes from the same prehistoric German source as does our bush. In the case of boscage it travelled through Frankish into Old French bos or bosc, both variants of bois, a wood, from which English acquired it about 1400. There is some learned disagreement here, however, with a few dictionaries pointing to a connection with the Latin boscum instead.
A dialect variant of bosc eventually transmogrified itself into the French bouquet, which was borrowed into English in the eighteenth century. It is also the source of the French word bocage for the landscape of small fields and hedgerows with tall trees and bushes characteristic of Normandy and Brittany. Our equally literary adjective bosky for areas covered with bushes or underwood derives directly from bosk, a dialect form of bush.
Boscage could also refer to ornamental plantations and be a poetical alternative for shrubbery. As an example, here is Thomas Carlyle in his monumental History of Friedrich II of Prussia, which ran to an indigestible 21 volumes: “We were all dancing in the fine saloons of Monbijou, with pretty intervals in the cool boscages and orangeries of the place”.