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The author, Jason Brown, achieves this effect using a standard drawImage with offsets. He uses a background image and a foreground image in columns offset in the y direction.
He created a canvas element in memory in which to draw the images, defined the height of the columns and set a fallSpeed transition for the columns.
You can read the original blog post here for a more detailed description including code snippets, and you can check out the codepen here.
Also, if you're interested in seeing how the original Doom developers implemented the screen-wipe effect, you can check out the relevant source code on GitHub.