Electrorheological fluids constitute a type of colloids that can vary their rheological characteristics upon the application of an electric field. The recently discovered giant electrorheological (GER) effect breaks the upper bound of the traditional ER effect, but a microscopic explanation is still lacking. By using molecular dynamics to simulate the urea-silicone oil mixture trapped in a nanocontact between two polarizable particles, we demonstrate that the electric field can induce the formation of aligned (urea) dipolar filaments that bridge the two boundaries of the nanoscale confinement. This phenomenon is explainable on the basis of a 3D to 1D crossover in urea molecules' microgeometry, realized through the confinement effect provided by the oil chains. The resulting electrical energy density yields an excellent account of the observed GER yield stress variation as a function of the electric field.