A static site generator lets you decouple the build phase of a site, form the actual hosting of the site.
For the final site, this means:
No backend: so all of your site can be hosted directly on a CDN, which will mean better performance than any dynamic system. Especially for global users.
No moving parts: this means all of the security issues with traditional dynamic sites just goes away.
No servers to scale: your CDN should be able to handle any kind of scale, so you never have to over-provision or setup complex auto-scaling strategies to handle large traffic spikes.
But it also means a ton to developers and their workflow, and this is often the primary reason to go in this direction.
Everything is in git: you don't have a binary blob of a database with all your actual content in it (and weird theme CSS, etc, in the case of something like Wordpress). This means full version control for all content, forever. This also means that all a freelancer needs to work on the site, is doing a git clone - and everything is in there. The local dev version of the site will be 100% identical to the site running in production, including all the content.
Everything is plain text: which means that as developers we can really easily work with all the tools we have at our disposal that's based on plain text. Make simple scripts for content migrations, do global search and replace, write markdown directly in our text editor of choice, etc, etc...
Fits a modern front-end workflow: today we all use transpilers, post-css, npm modules, task runners, etc. This is really hard to fit into a traditional dynamic site flow, but for static sites it's really easy to get an amazing front-end dev environement running with Browsersync, live reload, dev builds vs production builds, etc, etc..
Apart from that there's no a whole new ecosystem coming into existence with widgets like Disqus, Snipcard, Intercom, Optimizely and so on, that all works with static sites, as well as ton of API based services like Switftype, Contentful, RestDB, AWS Lambda, and so on, that makes it easy to integrate all kinds of advanced dynamic functionalities straight from your front-end code.
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