Basically textshots are images with quotes from articles. They highlight a specific part and are shared with the source url and sometimes a short commentary on Twitter or Facebook. They were first used by M. G. Siegler. Within a short time they became the standard way to link to noteworthy articles on Twitter.
I see two big problems with textshots in general. For one the images are totally useless without the tweet or the Facebook post. Some textshots include the complete or a cropped version of the source url but because it's an image, you can't click the link. The image on its own has no context.
The second problem is way bigger. Textshots are horrible for accessibility. It should be one of the big goals of developers to make websites and Apps as accessible as possible. Everyone should benefit from the web, no matter if they use their eyes, voice over or a braille display. Images have always been hard in regards to accessibility and textshots are no exception to that. All the technology developed over the years to help visually impaired people to consume content do not work for the text inside those images.
Linking to interesting articles isn't new. John Gruber has done it for years with link posts. Those are basic blog posts containing the link to the article and a plain text quote. All those links and quotes are perfectly accessible for everyone.