Are you trying to do a transition where you slide a layer off the screen and bring another layer in? If so, that's just key framing position or anchor point on two layers. If you want to also animate something like the X scale, then that is also just a matter of a few keyframes.
Are you new to AE? Do you understand the Basic Workflow?
Please do not double post the same question.
Just like I said before. Animate X scale and position. Set the anchor point of the left image to the left side. You can then use an expression or just generate keyframes for scale. When you flip to the next screen then animate position. Easy basic animation except for the bounce. Key framing the bounce so that it looks natural takes a lot of experience or an expression.
Google bounce expression after effects and find this. http://www.motion-graphics-exchange.com/after-effects/Wiggle-rubber-bounce-throw-inertia-e xpressions/4ad0f32a944ad
There are a bunch of other solutions.
There is no single click solution to this problem and it is not an effect, it is an animation.