When the Web was built in the late 1980s and early 1990s, it was designed along similar lines, with the assumption that browsers worked for the users and connected to servers over standard protocols. The technical term for a browser "user agent" reflects this.

In Mozilla's Web Vision (https://mozilla.org/en-US/about/webvision/full/) we put it like this:

Mozilla's Vision of the Web

Mozilla
But of course because the browser works for the user that means it can do things that the site doesn't like, such as block ads or let the user download pieces of content that the site would rather just be viewed but not saved.
WEI is a technology that provides "attestation" in which the site is able to verify certain facts about the software running on a remote machine, for instance "this user has an unmodified copy of Chrome.
In general, attestation works by having some piece of "trusted" hardware on the user's machine which asserts those facts. In this case "trusted" means trusted by the site, not the user, who after all might want to lie to the site.
The idea here is that if the site knows enough about what's on the user's computer then it can verify that that computer is behaving the way the site wants.
Aside from the obvious problem from the user's perspective that it allows the site to impose their preferences on the user, there is a bigger problem that it drastically limits the user's choices even on matters the site doesn't really care about.
There's no efficient way for the site to know whether a new piece of software behaves the way it wants, which means that a functional system will need to limit user choice to a relatively small number of pieces of vetted software on a relatively small number of platforms.
This isn't just a matter of "I want to use an unusual Web browser" though obviously that happens (and good luck if you want to run Linux!). Even a standard browser is very configurable, between settings and extensions/add-ons.
Each of these can affect the behavior of the product and the only really safe thing to do is to refuse to allow any unknown extensions, which will be most of them.
The point here is that even if the user isn't trying to do something that the site wouldn't like, the site is most likely to end up blocking it just to be safe, with the endgame here that the user's choices about what software to run and how it behaves are sharply limited.

This is the opposite of what it means to have an open Web.

https://educatedguesswork.org/posts/wei/

The endpoint of Web Environment Integrity is a closed Web

We had to destroy the Web in order to save it