As I understand it, the Li+ would be associated to the oxygen atom in the DMSO, leaving the sulfur atom with a (more or less) positive charge. Thereafter, the Polymer which is about to be dissolved could (e.g. with a -OH group) coordinate to the highly electrophilic sulfur atom and together with the leftover Br - be then solvated (as an ion pair or something similar) by the DMSO, which makes up over 95% of the solvent.
I would be happy to learn if there is evidence in the literature for this or if the mechanism for solvating polymers in mixtures of DMSO (or DMF, DMAc for that matter) and LiBr follows another concept. Thanks.