Stick-slip vibration of the bit in ultra-deep wells will seriously affect drilling potency, increase extra costs, and even cause drilling accidents. The study of the inner mechanism and external influencing factors of stick-slip vibration is of nice significance to the analysis on the suppression of stick-slip vibration. During this paper, Firstly, the interaction between the bit and rock stratum is analyzed through theoretical analysis. Based on the principle of J. Boussinesq elastic theory solution and Coulomb-Navier criterion, the damage process of the rock and the critical conditions for the incidence of stick-slip vibration of the bit were analyzed. Next, an analysis of the axial force variation of the drill column was carried out based on the establishment of an equivalent axial force variation model of the drill column torque transmission system, and the placement most strongly affected by the stick-slip vibration was located. Then, from the perspective of energy transformation of stick-slip vibration, it is analyzed that the undamped state of the bit at the instant of rock breaking is an important factor leading to the incidence of stick-slip vibration. Finally, it is verified through semi-physical experiments that adding damping to the bit at the instant of rock breaking can directly and effectively suppress the incidence of stick-slip vibration.