How would you create something like this with minimal effort?
Depends on what you mean by 'like this'.
The NYTimes hires talented interaction designers that create those presentations. They aren't from a template, but created based on the context of each story.
Broadly speaking, it's HTML5, CSS and JS.
Like most web design over the past 10-20 years, you either pick a template and live within the confines of the template, or you DIY by hacking away.
There are some tools out there that can help with parts of this, however. For example, Adobe's Edge Animate. I can't say how good it is. I'm skeptical, as I've never been impressed with software than auto-generates HTLM and CSS, but might be something worth looking at.
Side note: While those types of presentations are visually stunning, they aren't always the most usable way to present content. Often people just want to read the story...not have to scroll through pages of sliding bells and whistles. Something to consider.