About external core and internal core TiddlyWiki
A large part of TiddlyWiki's functionality is written in javascript as packaged as a TiddlyWiki plugin known as the "TiddlyWiki Core". This code is encoded as json into a single tiddler called $:/core. Because it's very large it accounts for more than 90% of the file size in a new TiddlyWiki.
Newer versions of TiddlyWiki support loading that core plugin externally as a separate javascript file. The advantage of this is that the TiddlyWiki file itself becomes much smaller, and because browsers can be instructed to cache the core javascript file, it generally doesn't need to be downloaded every time. Also, for Tiddlyhost in particular, it means loading and saving is much faster.
The disadvantage is that the core javascript file must be always available, so to use your TiddlyWiki offline you need to either rely on internet access to fetch the file, or you need to download a copy of the core javascript file and keep it locally with your TiddlyWiki file. There are also some other inconveniences related to upgrading; see more details here.
When creating a new site on Tiddlyhost, you can choose the "TiddlyWiki (external core)" option to create an external core TiddlyWiki. Note that the "actions" menu for this type of site gives you different download options for using your external core TiddlyWiki file locally.
Personally I prefer to use external core TiddlyWikis on Tiddlyhost because they're faster to load and save, but the default option is still internal core since that is better understood and better supported in the TiddlyWiki community, and likely to be less confusing for new users.