We report on new insights into the relationships between structure and activity of glassy carbon (GC), as a model material for electrocatalyst support, during its anodization in acid solution. Our investigation strongly confirms the role of CFGs in promotion of Pt activity by the "spill-over" effect related to CO(ads) for methanol electrooxidation (MEO) on a carbon-supported Pt catalyst. Combined analysis of voltammetric and impedance behaviour as well as changes in GC surface morphology induced by intensification of anodizing conditions reveal an intrinsic influence of the carbon functionalization and the structure of a graphene oxide (GO) layer on the electrical and electrocatalytic properties of activated GC. Although GO continuously grows during anodization, it structurally changes from being a graphite inter-layer within graphite ribbons toward a continuous GO surface layer that deteriorates the native structure of GC. As a consequence of the increased distance between GO-spaced graphite layers, the GC conductivity decreases until the case of profound GO exfoliation under drastic anodizing conditions. This exposes the native, yet abundantly functionalized, GC texture. While GC capacitance continuously increases with intensification of anodizing conditions, the surface nano-roughness and GO resistance reach the highest values at modest anodizing conditions, and then decrease upon drastic anodization due to the onset of GO exfoliation. We found for the first time that the activity of a GC-supported Pt catalyst in MEO, as one of the promising half-reactions in polymer electrolyte fuel cells, strictly follows the changes in GC nano-roughness and GO-induced GC resistance. The highest GC/Pt MEO activity is reached when optimal distance between graphite layers and optimal degree of GC functionalization bring the highest amount of CFGs into intimate contact with the Pt surface. This confirms the promoting role of CFGs in MEO catalysis.