This is an easy one. When you have your text, go to the text animate properties, and insert a position attribute. Keyframe your range selector start point, and then put the value in the position that you want it to change. Under advanced change your ease options or make an easy ease on the keyframes and in the graph editor manually change it.
Next precompose, then create a mask to mask out the bottom position, feather it a bit, and then play the animation. The text should raise up just like that add.
Instead of precomposing your text, you could also make a new solid (or square shape layer) and drag the edge to where you want the text to start appearing (not covering your text). Make sure it's the layer above your text layer and set your text layer to Inverted Alpha Matte. The solid will become invisible and this will make the text appear wherever the solid isn't. I like this method because it allows you to edit your text without making a new precomp. Then you can see how it looks in the comp without going back and forth. It's really just a matter of preference in most cases.