What Are Externally Hosted Scripts, Images And Stylesheets?
Externally hosted scripts, images and stylesheets are used by your shop but not hosted by your shop.
For example, a self-hosted script might be https://your-shop-url.com/slideshow.js. This slideshow.js script is hosted in your shop and you in most cases have full control over it.
Now if a developer runs https://amazing-slideshow-scripts.com and lets others link to her scripts, your shop can load the script directly from https://amazing-slideshow-scripts.com/slideshow.js instead of hosting it yourself.
Direct linking to externally host scripts, images and stylesheets can have a few benefits. First, if the external site is ran by a company like Google you can be sure its servers are fast and spread across the world. Users may have these resources 'cached' or saved to their computer if they've visited another site that uses the same slideshow for example. They don't need to be downloaded again.
You may also benefit from upgrades straight away when the developer updates their code. You don't need to check for updates, download the files, install them in your shop and update and links- the developer will simply update her external site and you get the new version.
With Shopify this can be an issue if you use Shopify apps. Some install their code in your theme, but some link out to scripts, images and stylesheets hosted on their own servers. These are unfortunately not able to be speed optimized.
On a positive note though, many Shopify app developers are responsive to feedback. So if something is flagged as a speed issue to them they may fix it on their servers. Our recommendation is to try reaching out. Plug in Speed users feel free to CC firstname.lastname@example.org and we can weigh in on any technical questions.