It wasn't long after Facebook acquired Twitter that the blue bird blocked Instagram photos from showing up in your Twitter timeline. That was an understandable move, but an annoying one, since it forced users to click on any Instagram link and view it in a browser just to see the photo.
The recipe works by sending Instagram photos directly to Twitter's own picture service, making it appear (to Twitter) as if you simply uploaded the photo via Twitter itself. Thus, your photos will show up inline, saving your followers the hassle of opening their browser:
By default, the recipe will include a "via Instagram" snippet, but you can change that in the recipe itself if you want to save the extra characters.
Obviously this workaround won't do anything to appease your frustration, since you'll still have to open Instagram links in your browser to view the photos of anyone who doesn't use this recipe. Still, it's one less annoyance your followers have to deal with (and hey, you could always share this article on Twitter; the more the recipe is used, the less widespread the nuisance becomes).