I've seen a titles which tell that it's a nightmare for those companies (Google, Facebook, Twitter). But they're pretty vague on it. What exactly does it do to them?
The executive order is fundamentally about Section 230 of the Communications Decency Act of 1996.
Section 230 says that "No provider or user of an interactive computer service shall be treated as the publisher or speaker of any information provided by another information content provider" (47 U.S.C. § 230).
The distinction of "publisher or speaker" here is important. Summarizing from DMLP: A publisher such as a book or newspaper is liable for any illegal or defamatory content they publish. "The theory behind this "publisher" liability is that a publisher has the knowledge, opportunity, and ability to exercise editorial control over the content of its publications."
Also according to the site, "Courts have consistently held that exercising traditional editorial functions over user-submitted content, such as deciding whether to publish, remove, or edit material, is immunized under Section 230. As one moves farther away from these basic functions, immunity may still exist, but the analysis becomes more fact-specific."
From the Executive Order itself: (As an aside the text is highly readable and I suggest reading it yourself)
Section 230(c) was designed to address early court decisions holding that, if an online platform restricted access to some content posted by others, it would thereby become a “publisher” of all the content posted on its site for purposes of torts such as defamation. As the title of section 230(c) makes clear, the provision provides limited liability “protection” to a provider of an interactive computer service (such as an online platform) that engages in “‘Good Samaritan’ blocking” of harmful content.
The question is now to what degree social networks are performing "editorial functions" and to what extent this is in "good faith", such as restricting access to content that is “obscene, lewd, lascivious, filthy, excessively violent, harassing or otherwise objectionable”. For example, according to Trump's Executive Order, "Twitter now selectively decides to place a warning label on certain tweets in a manner that clearly reflects political bias."
Here is my selective summary of the sections and actions described.
Sec. 3: Each executive department and agency will review its funding of online platforms problematic vehicles "for government speech due to viewpoint discrimination, deception to consumers, or other bad practices."
(a) It is the policy of the United States that large online platforms, such as Twitter and Facebook, as the critical means of promoting the free flow of speech and ideas today, should not restrict protected speech.
(c) The FTC shall consider taking action, as appropriate and consistent with applicable law, to prohibit unfair or deceptive acts or practices in or affecting commerce, pursuant to section 45 of title 15, United States Code. Such unfair or deceptive acts or practice may include practices by entities covered by section 230 that restrict speech in ways that do not align with those entities’ public representations about those practices.
(d) For large online platforms that are vast arenas for public debate, including the social media platform Twitter, the FTC shall also, consistent with its legal authority, consider whether complaints allege violations of law that implicate the policies set forth in section 4(a) of this order. The FTC shall consider developing a report describing such complaints and making the report publicly available, consistent with applicable law.
It should be noted that in Knight First Amendment Institute v. Trump, the United States District Court for the Southern District of New York ruled that the @realDonaldTrump account constitutes a public forum and blocking access to it is a violation of First Amendment rights.
Here is the full text of the order itself.
Trump’s directive now could set the stage for federal regulators to write new rules and issue new punishments for companies deemed to exhibit political bias. Depending on how the order is carried out, it poses the potential for wide-ranging consequences for a much broader segment of the Internet beyond just the social media giants, potentially affecting every website, app or service where users congregate online with new liability for the content on their platform.
Lawfare blog has a more detailed analysis, arguing that most of the main requirements will not stand up legally, but will still have the desired effect putting pressure on social media companies "that do not kowtow to Trump."