What is the legal process whereby the proposed bans of RT and Sputnik are supposed to be put into practice? Why does the European Council think it has the right to force ISPs to censor websites?
From comments, Which laws allow the European Council to order ISPs to violate net neutrality? is a more legally answerable question than "Why does the European Council think it has the right to force ISPs to censor websites?".
EU law on net neutrality is contained in the "Open Internet regulation", 2015/2120, principally its Article 3, in which we read (clause 3):
Providers of internet access services shall treat all traffic equally, when providing internet access services, without discrimination, restriction or interference, and irrespective of the sender and receiver, the content accessed or distributed, the applications or services used or provided, or the terminal equipment used.
This goes on to add, with irrelevant words omitted:
Providers of internet access services ... shall not block ... specific content ... except as necessary, and only for as long as necessary, in order to comply with Union legislative acts, or national legislation that complies with Union law, to which the provider of internet access services is subject, or with measures that comply with Union law giving effect to such Union legislative acts or national legislation, including with orders by courts or public authorities vested with relevant powers.
This is a long-winded way of saying that other provisions of national or EU law can require content to be blocked. Paragraph 13 of the preamble explains it in more normal language - the net neutrality provision is not intended to give carte blanche to the publication of material which would otherwise be unlawful.
In the case of RT and Sputnik, Regulation 2022/350 is an example of such a law, saying:
It shall be prohibited for operators to broadcast or to enable, facilitate or otherwise contribute to broadcast, any content by the legal persons, entities or bodies listed in Annex XV, including through transmission or distribution by any means such as cable, satellite, IP-TV, internet service providers, internet video-sharing platforms or applications, whether new or pre-installed.
and also "This Regulation shall be binding in its entirety and directly applicable in all Member States." The Council is allowed to make such regulations by virtue of Article 215 of the Treaty on the Functioning of the European Union, on "restrictive measures" against "natural or legal persons and groups or non-State entities", with certain procedural prior conditions too lengthy to discuss in detail.
Therefore, if your country is a party to that treaty and has arranged for its direct effect in domestic law, then Regulation 2022/350 is binding and supersedes the general rules of Regulation 2015/2120.
By general principles of the primacy of EU law, the ban also overrides any contrary provisions of national law, including perhaps even a national Constitution. Since you are in Germany: under the "Solange" principle accepted by the Bundesverfassungsgericht, domestic courts would not review EU law by the standards of the human rights provisions in the Grundgesetz, since the EU institutions are held to be capable of providing their own parallel safeguard of essentially the same rights. This Regulation could therefore be challenged on the grounds of Charter/Convention rights, or compatibility with the Treaties, but not because of conflict with a national law on net neutrality (if there was one).
The preamble of the recent Regulations does acknowledge the human rights angle and claims to be compatible with the Charter of Fundamental Rights. While the ban does interfere with the freedom of expression, that right is subject to certain limits; and ECHR Article 16 explicitly reserves "imposing restrictions on the political activity of aliens".
One thing the EU is good at is cross-referencing its legislation. The council decision cited in the question mentioned in the comments begins:
Having regard to the Treaty on the Functioning of the European Union, and in particular Article 29 thereof, ...
That article reads
The Council shall adopt decisions which shall define the approach of the Union to a particular matter of a geographical or thematic nature. Member States shall ensure that their national policies conform to the Union positions.
The decision goes on to cite several other instruments in its recitals before amending a decision from 2014 that no doubt cites still more instruments. The key justification is probably that in recital 8:
Those propaganda actions have been channelled through a number of media outlets under the permanent direct or indirect control of the leadership of the Russian Federation. Such actions constitute a significant and direct threat to the Union’s public order and security.
The authority flows from Article 29, and more broadly Chapter 2, concerning "common foreign and security policy," of the Treaty on European Union.