In the context of acoustics, MicroPerforated Plates (MPP) are commonly used as sound absorbers. Such systems involve exchanges in the viscous and thermal boundary layers in the microperforations, which can result in sound absorption. It was also observed that fluid-solid interactions in the microperforations along with the previous effects can lead to additional structural damping. However, little is known about the vibration dynamics of MPPs and their damping capabilities have never been investigated in details. This additional damping could be of high interest in the design of new low-cost, simple, and purely passive solutions targeting the mitigation of structural vibration. In the present work, an analytical model is developed by identifying a finite-size MPP as a porous plate and using an alternative form of the Biot theory of poroelasticity. In this model, the MPP is considered as an equivalent homogeneous plate. The equivalent mass density and Young's modulus of the homogenized model account for the presence of microperforations. Free and forced-response analyses are performed to investigate on the vibratory behavior of the system. It is shown that the added damping reaches a maximum at a characteristic frequency which only depends on the perforation parameters, and thus can be tuned easily. Also, the magnitude of the additional structural damping in the MPP increases as the characteristic frequency decreases. Furthermore, the damping mechanism is effective over a significant bandwidth around the characteristic frequency and is optimal when the first natural frequency of the MPP coincides with the characteristic frequency. The analytical model is validated by measurements on MPPs which confirm the significant damping increase in the low-frequency range.