Based on the first-principles of the density functional theory, the Gibbs free energies (△GH0) of the hydrogen adsorption on the 2H-phase molybdenum diselenide monolayer (MoSe2) with different active sites and hydrogen coverage rates are calculated. The calculated results reveal that several ideal adsorbed rates and sites are very close to those at thermoneutral state (△GH0~0). To compare their catalytic ability in the hydrogen evolution reaction (HER), the exchange current density (i0) as a function of △GH0 is calculated as a volcano curve. Two sites located at the top of volcano curve present higher exchange current densities than that of Pt catalyst. The charge transfers and the bonding details of the two edge-hydrogen-adsorptions (Mo edge and Se edge) are analyzed by the charge density difference and electronegativity as the associated structures and relative △GH0 are further explained. It is found that the localized charge transfer distributed uniformly between the hydrogen atoms and the adsorption sites can facilitate the catalytic ability of HER. For this reason, the catalytic ability of HER for the Se edge is more stable than that of Mo edge with less sensitivity to the absorption sites and hydrogen coverage rates. Based on the first-principles molecular dynamics (MD) simulation, finally, the influences of the thermal motion on the two kinds of structures of hydrogen adsorption at the higher temperature are explored, with the critical temperature for the hydrogen desorption as well as the atomistic dynamics discovered. It is worth mentioning that during the structure optimization and MD simulation, the edge deformation and reconstruction are discovered, respectively, which indicates that the ideal edge of MoSe2 may not be the most stable structure, which will change with the external conditions. This theoretic study reveals the atomistic mechanisms of the hydrogen adsorption and desorption of the single-layer 2H-phase MoSe2 at different temperatures, with the edge lattice deformation and reconstruction discovered, which can deepen our insights into the HER mechanisms near the edges of the 2H-phase MoSe2 at different temperatures and provide theoretic guidelines for designing the high-efficient and low-cost catalyst in the HER through tuning the MoSe2 edges.