This is using Github Pages and Jekyll.
The essays were copied over from the WordPress admin dashboard, which is still accessible. Unfortunately, I couldn't figure out how to make images work, so the images have been deleted.
The redirection is happening on the WordPress server in .htaccess. It redirects everything but admin and login to the Github Pages for this account.
You can edit the text on the homepage by editing index.md. That file is in Markdown format.
nyt is just literally copied from the nyt repo (twice).
The different sections (features, essays, etc) are all Jekyll collections. You can edit them or add new ones in _config.yml.
Each collection has a directory like _features or _blogs-marginal-utility. Each file in that directory is a post.
I already added the 3 newest features, the 3 newest essays, and the 3 newest blog entries. I had to add paragraph breaks and remove the images.
I tried to mirror the paths of the WordPress site so links still work.
The overall layouts are in _layouts: index.html for the index page and 404 page and posts.html for all the different kinds of posts.
The layouts include other bits of html from the _includes directory: header.html, footer.html, recent.html (described above), and linkassets.html (the references to the stylesheets).