Preface: it has been a minute since I studied these systems in detail, this is the jist of it. Vitamine c is a redox active molecule existing in an equilibrium between reduced and oxidized forms tending to favor the reduced form. Ascorbic acid (reduced vitamin c) gets oxidized by taking on a single electron from a radical (molecule with an unpaired electron) to form oxidized vitamin c. If it is exposed to oxygen or other oxidants for long periods of time it will exist largely in the oxidized form and end up being a prooxidant rather than antioxidant in order to re-establish the equilibrium. Your body is able to reduce oxidized vitamin c via various intracellular antioxidants (glutathione etc) but that initial exposure will produce some ROS.
Also in cell culture, there's often a high concentration of free iron floating around (5mM) which is totally unnatural. The free iron/vitamin c combo is able to create a Fenton reaction and produce huge amounts of ROS.