According to Blogger Official Blog for Known Issues, animated files does not remain animated when hosted on blogger because
Picasa doesn’t support animated images and the Blogger-hosted photos are stored in Picasa Web Albums.
A solution to this problem is to use some other Image hosting solution such as Tinypic, Photobucket, Pictures.pk,etc.
Just upload your animated photo on any of these hosting sites and get the html code for the Image. Now, go to Edit HTML while writing the post and paste that HTML code for animated image to your post. You will get an Animated Image in your Blogger Blog Post. Here is an Example below of an Animated Image, Enjoy!