Let's visualize OVER in our minds. It is...
A girl combing her hair
A hand turning the pages of a book
A bridge across a river
A glance from top to bottom and then back again
A child rolling down a hill
Poetic, isn't it?
Go back to the examples again, and look for this half circle shape:
A girl brushing her hair (She brushed through her hair over and over, or she combed over her care carefully, looking for nits).
A hand turning pages: (He turned over each page carefully
Moving over a bridge - in this case literally:
Glancing from top to bottom and/or from side to side (He looked over his surroundings carefully).
A child rolling down the hill - or in this case, a panda (The panda rolled over and over down the hill)
Often this motion can be repetitive.
To do something over means to do it again...
and to start over means to go back to the beginning.
Think of over as moving or changing from point A to point B. There's no continuation (as on implies), or worry about what the path between start and finish looks like, and you can start again as many times as you like (do over!). But when you're finally, permanently at point B, you are DONE! (It's finally OVER!)
Here are a few common phrases that access this meaning of the word OVER:
Come over here! (from where you are to where I am)
move over (please!) (move a little from here to there)
Ok, hand it over... (from you to me)
"Over" additionally implies a sense of control and authority. More on this sense of OVER in another post.