Manual:$wgAllowExternalImages

Whether to allow inline images hosted on external websites, sometimes called hotlinking.

Details
When set to true, users can bring in an inline image from an external URL. They do this simply by sticking the full URL into the wiki text (without any other surrounding markup) That is the direct URL to an image resource on another website.

When set to false, the full URL appears as a text link to the image.

Example:
Using wikitext:  http://en.wikipedia.org/upload/b/bc/Wiki.png 

Result: http://en.wikipedia.org/upload/b/bc/Wiki.png

Thumbnails of external images
For resizing images in mediawiki they need to have a row in image table of database containing dimensions and other information of image so you can't resize external images.

If those images have been uploaded in another mediawiki or if they are somewhere in your site and you want to let mediawiki make thumbnail of them you can set $wgUseSharedUploads to true and set $wgSharedUploadPath and $wgSharedUploadDBname instead of enabling external images.

Why allow external images?
Users who know how to get the direct URL of an image, may find this a very useful way of adding illustrations to wiki articles, using image resources available elsewhere on the web.

Without this feature, users are forced to upload any image which they want to embed directly on an article. Upload is (in many ways) more fiddly and time consuming. It also uses disk space and bandwidth on the wiki webserver of course. There are many legitmate uses of inline images.

Why disallow external images?
Inline external images are disallowed on across many wikimedia projects, and MediaWiki has AllowExternalImages set to false by default since version 1.6.9 Reasons for this.... To prevent (or reduce the impact of) certain types of vandalism. People can still upload offensive images to your wiki, but this option will prevent people inline linking to offensive images, which might make that kind of vandalism less prevalent. If you do allow external images, then removing abusive image links is always a matter of a simple wiki edit (followed by whatever conflict resolution/banning action you wish to take)

To prevent "bandwidth theft" from other sites. A wiki administrator might wish to avoid being accused of bandwidth theft due to inline linking.

This is only of concern in cases where your wiki is linking to a web server/host belonging to somebody else, and that person was not involved or consulted in setting up the link (a wiki edit) In that situation the person might object to such links. Of course removing the link is as simple as editing the wiki, but it should be noted that inline linking in this kind of situation is widely regarded as bad netiquette (even if you are very happy to remove on demand).

In many other situations, bandwidth theft is not a concern:
 * There are some websites which explicitly encourage inline linking to images they host (note that sometimes terms & conditions are attached to inline linking e.g. they require a textual link/acknowledgement)
 * A contributor might decide that images they are hosting on their own website could appear on a wiki page, and so they set up the inline link themselves.
 * Often it might be useful to allow inline linking to images hosted on the very same webserver as the wiki (which are just not uploaded into the wiki), or images hosted as part of the same overall project/organisation.

To ensure you are hosting all resources. You might wish to disallow inline linking in order to ensure that all images resources appearing on your wiki, are available from your wiki. If you have good hosting bandwidth, you might want to make full use of this and avoid any slowdown in browsing experience due to images being fetched from elsewhere. You might also want to ensure you are gathering all the image resources locally, in order to offer up all your content as a download.