In general this sort of transition is called a wipe. I might call it a "layered horizontal wipe". It's generally done by layering two images and then animating a clip path that's applied to the top image. Or you could fix the top image (so that it doesn't resize its dimensions) and animate its width.
It made me think of a lot of the transitions in Star Wars ...
You could use Illustrator CC's Rounded Corner Widget functionality.
Draw a shape without curves, then select only the corner widgets you want to round by Shift+clicking them to make a multiple selection, then click and drag them to make the curves.