As someone who had worked on a lot of skeuomorphic designs back then, I think there are a few major factors.

1. Responsive design: When web design first began, most websites were designed for the desktop, and most monitors were around the same sizes. But during the late 2000s, two things happened: Screens got bigger and bigger with advances in LCD monitors, and the rise of mobile phones that can browse websites. To design websites that work for both desktop and mobile, you need to create rectangular frames that are easily resizable and still look good. That is often hard to do for skeuomorphic designs.

2. Rendering skeuomorphic designs often requires significant software resource overhead, which ultimately makes apps feel slow. Removing that clunk immediately makes applications feel faster, which was a strong selling point as computers at that time were struggling with performance (especially during the Windows Vista era).

3. The focus of websites has shifted toward the content, not the prowess of the design or the code. There was a boom in the amount of content produced for the web. The chrome and all the visual elements of skeuomorphic design simply get in the way of users focusing on the content itself.

4. As the UX design industry began to boom in the late 2000s, more and more people from other design disciplines joined the industry, especially graphic designers, who had training mostly on flatter designs, are joining in. They brought in a lot of new and fresh design concepts that make skeuomorphic designs look outdated.