What you have is an archive.
Linux has a massive manual with nearly every command right at your fingertips. These are called the man pages. To use man command, and it will give you everything on how to use it. In your case, you want tar to unarchive your file. To see how to do this, man 1 tar, or simply man tar will show you all of the options for archiving and unarchiving. The tar command is one that you'll use a lot, so it is helpful to know some of these and know where the man pages are.
Anyway, the version of libwebp that I found on google was 0.4.0, so once I unarchive it, use cd libwebp-0.4.0 to change to that directory.
Then according to the readme (less README), it says to install use:
./configure
make
make install
you may need to put sudo in front of the last command and it should install the library.
Now, you didn't say which distribution of Linux you are using, but I wouldn't be too surprised if the library is already installed.
If you don't feel comfortable building software, yet, I would recommend looking at your distribution's repository.
To install in ubuntu from the repo:
sudo apt-get install libwebp
Once it is installed, you can use cwebp to compress your image. dwebp will decode it, vwebp will let you view.
Once you have your webp images, you want webpmux to create your animation. I haven't fully figured out how to use it effectively, but to create a webp animation you can:
The use format according to the man pages is:
webpmux -frame file_i.webp +di[+xi+yi[+mi[bi]]]
file_i is the ith frame. 'di' is the pause duration, so how long on that frame.
'xi', and 'yi' are the x and y offset for the frame.
'mi' is the how the frame is to be disposed. 0 = None, 1 = Background
'bi' is the blending method for the frame: +b = Blend -b = No Blend
The man pages can describe it in more depth. I recommend giving them a read. To see them use man webpmux, man cwebp, and man dwebp.
You will also find an example of how to use each in the man pages.
Good luck.
Personally, unless you have a reason for using the webp format, I'd use another animation video encoding format (GIF, webm, etc...) which can be made with graphical tools like openshot or gimp.