Predicting torque before you need it
At its core, a dynamics model is an equation, or set of equations, that takes the robot's current position and velocity at every joint, along with the acceleration you want it to achieve, and outputs the torque needed at each joint to make that happen. Nothing more mysterious than that, it's a function that maps 'where you are and where you want to go' to 'how hard each motor needs to push'.
The value of having this model is that it lets you predict rather than merely react. Without one, a robot's controller would have to discover the right torque by trial and error, pushing a little, seeing what happens, adjusting. With an accurate dynamics model, the robot can compute in advance roughly how much torque a motor will need for an upcoming motion, before that motion even starts.