There are a few websites that I stumble upon now and then that have animated intros like the one here: www.monotwo.com
How do I do that in WordPress?
And how would I integrate this into a customs theme?
You can try this.
or you can research follow these keywords to know more detail.
Monotwo uses SVG + JS for their animation, exported from Adobe CS (probably After Effects or Animate). See Creating a vector animation for the web in a post-Flash world for more info about this process.
You could also animate using Adobe Illustrator or Photoshop, Sketch, Hype - search for "vector animation" to find tools and techniques that fit you.
How do you do this in WordPress, and in a custom theme? Place your exported code in a static front page by using