Amendments to the US Constitution are part of the US Constitution, and this includes the so-called Bill of Rights. The doctrine of applying parts of the US Constitution to states, known as incorporation, comes from the 14th amendment:
No State shall make or enforce any law which shall abridge the
privileges or immunities of citizens of the United States; nor shall
any State deprive any person of life, liberty, or property, without
due process of law; nor deny to any person within its jurisdiction the
equal protection of the laws.
The scope of the doctrine of incorporation is logically limited to amendments because the original articles of the constitution state how the federal government is run, and does not guarantee any rights or privileges to individuals. Amendments 1,2,4,8 are fully incorporated against states, 5 and 6 are partially incorporated, third and 7th are not incorporated, and for 9th and 10th there is no ruling. This arrangement derives from decisions by the Supreme Court, interpreting the 14th Amendment and the concept of "due process".
If a state were to passe a law forbidding criticism of the governor (violating the First Amendment), an individual could sue to have the law found unconstitutional. Because, under the 14th Amendment, "The Congress shall have power to enforce, by appropriate legislation, the provisions of this article", Congress can and has passed a law prohibiting violation of civil rights: 42 USC 1983. Since the state in this hypothetical scenario has violated federal law, the federal government (Dept. of Justice) has standing to sue the state for
violating federal law.