N-Bromosuccinimide is an oxidising reagent and essentially supplies a small amount of Bromine radicals. These would oxidise vitamin C or ascorbic acid.
Ascorbic acid is an ene-diol which, with a radical, forms the corresponding H0-C=C-O. radical which is stabilised (this is how Vitamine C prevents bad radicals from oxidising our cells). This radical would itself ultimately get further oxidised with bromine (not much of that in the body!!) So, overall, ascorbic acid is oxidised by bromine.