That is some text and graphics precomped, layered and moves into place with a couple of keyframes. Pretty simple animation, which leads me to believe you are probably quite new to AE. Take a look at this set of TUTORIALS from VideoCopilot. They will quickly run you through the basics of text, keyframing, transformations, everything you would need to make that effect.
That being said, to create that, drag your footage on to the create new Comp button and in the projects panel rename it "Final".
Press "Cntl + N" to create a new comp and call it "Message 1", use the text tool, shape layers, graphics...to design your phone message.
Go to "Final" and move the Current Time Indicator(CTI) to where the first message should appear and drop "Message 1" into the comp. Press "S" to change the scale of the message. If you would like it to move a little like the first message does(In example video), press "P", move the CTI to when the movement should finish and press the little stop watch icon(this sets your ending keyframe). Now press "I" to move to the beginning of your "Message 1" layer and either physically grab the layer and move it slightly, or use the X coordinates and slide them up or down to get the layer to move to it's starting location. A new keyframe will automatically be generated.
Press "0" on the number keypad and your frames will be cached to RAM(you will see a green progress bar) and when the RAM cache is full it will play back your animation. You can change the starting and ending points of your RAM preview by pressing "B" and "N". Repeat these steps for all the messages.
"I have not failed 700 times. I have succeeded in proving that those 700 ways will not work. When I have eliminated the ways that will not work, I will find the way that will work."
---THOMAS EDISON on inventing the light bulb.