I want to describe this action. How should I describe this action?
I suspect your question will be closed as it leans to asking what to write (I'm not completely sure though, so no close vote from me yet), but I'll give you some of my ideas anyway.
The GIF shows the reader not really comprehending what the book is saying (unless they are incredibly fast at reading), rather they are jumping across pages, perhaps to find a specific section, or just out of boredom.
I would describe that action with these words:
a) flipping through pages
b) skimming through pages
d) surfing through the book
e) scanning the book
f) thumbing through pages
And then I would put them into sentences like:
a) Juliet carelessly flipped through the pages. "Ugh," she groaned, "this was such a stupid assignment."
b) Skimming through the pages, George finally found something of interest: Mart's address.
c) Hank sighed. Another day, another book, it was getting old. He quickly browsed through the dusty collection of fairy tales before archiving it in the library's catalog.
d) Maya surfed through her favorite book, she had read it hundreds of times, but it was the next best thing to do in her boredom.
e) Jack scanned the manuscript for errors only to find that the book was free of mistakes. Disappointed, he forwarded the attachment to a different beta reader.
f) Thumbing through her yearbook, Jessica had the intention of finding her face to see if her photo turned out right.
I would describe the action in the gif as riffling.
Riffle - gerund or present participle: riffling
turn over something, especially the pages of a book, quickly and casually. "he riffled through the pages"
This implies a casual action, if you want a similar process but implied urgency then you could use the similarly spelled rifling
Rifle - gerund or present participle: rifling
search through something in a hurried way in order to find or steal something. "she rifled through the pages"
Although I think rifling would be more appropriate for loose pages on a desk rather than in a book.
Well, what your animation shows is someone riffling pages as one other answer states.
But an alternative understanding relates to the older flip book technique (see https://en.wikipedia.org/wiki/Flip_book) that animated *.gif files digitally reprise/simulate.
So, you have choices, 'flip' or 'riffle' through. I prefer 'riffle through' as it implies going rapidly; whereas, to me, 'flip through' implies a casual perusal maybe seeking/hunting through to a page you stopped off at some time ago and regretfully forgot to place a bookmark.