Easy to explain. First, understand that there are two computers involved in any internet transaction: the home user's "client" computer, which sits on her desk at home or office, and the website's "server" computer, which can be anywhere in the world (and often is). The client-side computer is called the "front end" and the server-side computer is called the "back end".
In the old way of doing things, a website editor has to run a bunch of programs on her home computer in order to create a webpage. She has to use one program to edit photos, another program to convert documents to html, another program to upload them, yet another program (an internet browser) to view them online, and a handful of other specialized programs for various webmastering tasks. This requires installing a lot of expensive software on one's own computer, learning how to use it, and keeping up with regular upgrades. Other than storing the files, the server computer does almost nothing - all the work is done on the client-side computer.
With a CMS, the only program a user has to use is the internet browser like Internet Explorer or Netscape, and everything else is done on the server. After preparing the HTML, an editor uses the browser to upload and edit documents and the server software takes care of the rest. In other words, it's almost a complete reversal of the old method! This puts all the burden on the website programmer, and relieves the home user of any technical knowledge.
Here's the actual process:
Content Management Systems represent a wholly new paradigm for using computers. Home users have taken two decades to get used to the idea of a file system, where data is stored in discrete documents, each with a name and a location in a directory. This paradigm was intended to be a close facsimile to a real office, with desktops, files, folders, and so forth. In a physics analogy, this is a Newtonian virtual world.
CMSs will lead users to accept a new paradigm, one in which the filesystem has been discarded in favor of a database back end which is interpreted by PHP and the HTML interface is created on-the-fly. In this paradigm data is nebulous and without fixed form. In our physics analogy it is a quantum reality, where content is only given shape in the process of interacting with it. It is thus an extremely flexible virtual world, where structure is created by interpretation and can thus be infinitely expanded, shaped, and delivered.