How to specify a background image in a view so that it doesn't get removed?
To understand why this (= the style being removed) is happening, refer to this comment in the views issue queue:


  Rewrite uses filter_xss_admin and allows only the tags listed there.


It also includes a possible solution for it, which is like so:


  If you need more than this, then you need to use a template where you have complete control. This ensures that data entered via the UI is safe.


For an alternative solution (instead of using a template), you may want to consider the Background Images Formatter module. Here is an excerpt from its project page:


  This module provides an image formatter who allows you to set an image in background of a div. The images are from a field of an entity and not from a configuration page or sth else, so it's very easy to setup and manage.