How to implement

First, create the basic UI of the screen how we want to make it look.

Within the scaffold, use Stack Widget to place the cards on top of one other and position them by wrapping it in Positioned Widget to give the card deck effect.In the Stack of cards, the card at front is only the active one showing all the animation. Other cards are just dummy. So we need to add all logics and animation to the active card only.Here is my card design.You can design your card and use it.

In the initState() initialise the controller:

_buttonController = new AnimationController( duration: new Duration(milliseconds: 1000), vsync: this);

Animating any object is basically your perception towards it. Here if we closely observe the active card when animation starts, it is just moving slightly up, starts to rotate and while rotating it is moving in the specified direction(left or right).