A systematic theoretical investigation is performed for electrostatic potential of mean force (EPMF) between two similarly charged rods (modeling DNA) immersed in a primitive model electrolyte solution. Two scientific anomalies are disclosed: (i) although a like-charge attraction (LCA) generally becomes stronger with bulk electrolyte concentration, the opposite effect unexpectedly occurs if the two rod surfaces involved are sufficiently charged and (2) contrary to what is often asserted, that the presence of multivalent counterion is necessary to induce the LCA, it is found that the univalent counterion induces the LCA solely only if bulk electrolyte concentration rises sufficiently and the rod surface charge quantities are high. On the basis of the system energetics calculated first by a classical density functional theory in three-dimensional space, a hydrogen-bonding style mechanism is advanced to reveal the origin of the LCA, and by appealing to fairly common-sense concepts such as bond energy, bond length, number of hydrogen bonds formed, and counterion single-layer saturation adsorption capacity, the present mechanism successfully explains the scientific anomalies and effects of counterion and co-ion diameters in eliciting the LCA first investigated in this work. To add weight to the hydrogen-bonding style mechanism, a theoretical investigation is further performed regarding the effects of the rod surface charge density, co-ion valence, relative permittivity of the medium, temperature, nonelectrostatic interion interactions, and rod diameter in modifying the EPMF, and several novel phenomena are first confirmed, which is self-consistently explained by the present hydrogen-bonding style mechanism.