Based on the stiffness theory, the stiffness difference and energy supply of surrounding rocks are important inducing factors for strain burst. However, due to the lack of multi-level low stiffness testing machine, there are few experimental studies on rock failure mechanisms under the influence of loading stiffness. This paper introduced a self-developed rock testing system with changeable stiffness. The main testing machine of this system is a combined structure of inner and outer frame. The change in the loading stiffness is achieved by using a stiffness servo control system to control the energy accumulation in the inner frame. The tests of the sandstone specimens under three different loading stiffnesses in the testing system showed that the loading stiffness did not significantly affect either the uniaxial compressive strength or the Young’s modulus of the rock. However, the post-peak stress-strain curve of the rock became smoother and steeper when the loading stiffness was decreased, and the stress drop rate increased. It is shown that the stress drop rate had a power function correlation with the loading stiffness. After the peak load, the inner frame of the testing system rebounded several times at high speeds. The magnitude of the instantaneous rebound speed and the rebound duration increased with the decrease of loading stiffness. Both the mean rebound velocity and the total rebound deformation had power function correlations with the loading stiffness. The loading stiffness has limited effect on the energy storage and energy dissipation characteristics of the specimen, while the released energy of the testing machine increases as the loading stiffness decreases, which is the key factor affecting the failure mode of the specimen. According to the test results, a rock burst mechanical model considering the stiffness is established, and the energy supply mechanism of rock burst is quantitatively explained.