how to smoothly increase a video in picture of picture mode
What you're trying to do is easily achieved with after-effects. iMovie really isn't made for motion graphics.
What you want to do is the following:


Create a composition from your footage
Drag the pip video above the main video.
Press "s" while having the pip video selected to bring up the scaling.
Scale the video down and drag it to the position you'd like.
Now press the stopwatch on the scaling (this will create a keyframe).
Go forward in the timeline and enter 100% in the scaling, which will make it full-screen again.
Do steps 5-6 with the position (you can bring it up by pressing "p"), except for entering 100%, you'll right-click the position and choose "reset". This will center the now full-scale pip video perfectly.


Notes: It is important that the first keyframe from scaling and position are on the same frame, and the last  keyframe of position and scaling must also be on the same frame.

If you look at the animation, it might look a bit "linear". To make it smoother, you can select all four keyframes you have created, right-click and choose "easy-ease". This will interpolate your keyframes in a smoother way.