The mechanical connection between the paddle assembly and the face plate is called the sear. There are a lot of ways to design the target sear. Below are three common designs. This is by no means the only way to design it, however it will give you the general idea of how it can work. This design uses the weight of the paddle falling against the rear bar to make the face fall rearward.
Gravity is the main force making the face fall.
The sear is engaged only on a miss. The taret face is moves backward and locks up against the paddle arm, thus the target face isn't allowed to fall. This design uses a sping attached to the target face to make the face fall rearward. This makes the face fall very fast and flat. The paddle arm is connected to the target face via a small arm.