Of course you're correct that the BoR does say 'Congress' ... but ... the Law of the Land has changed over time.
For a decent history of how it has evolved, here's a well-written history:
http://faculty.smu.edu/jkobylka/supremecourt/Nationalization_BoRs.pdf
Basically, the 14th Amendment has been interpreted to mean that States also have to adhere to the Constitution.
A relevant passage on the 1A portion of this process:
Whether the "assumption" in the Gitlow case was a reality remained unclear until 1931. In that year, the Court decided Stromberg v. California and Near v. Minnesota and ruled squarely that the due process clause guaranteed both freedom of speech (Stromberg) and freedom of the press (Near).
"It is no longer open to doubt," Chief Justice Charles Evans Hughes wrote in the Near case, "that the liberty of the press and of speech is within the liberty safeguarded by the due process clause of the Fourteenth Amendment. It was found impossible to conclude that this essential liberty of the citizen was left unprotected by the general guaranty of fundamental rights of person and property."
After the freedoms of speech and press had been nationalized, the question remained about other rights in the First Amendment. Did they come under the Court's definition in the Twining case, as fundamental principles of liberty and justice which inhere "in the very idea of free government" and which are "the inalienable right of a citizen to such a government"?
The Court addressed these questions in short order. The right of assembly and petition, the free exercise of religion and the prohibition of an establishment of religion were held to be applicable to the states via the due process clause in the Court's decisions in DeJonge v. Oregon (1937), Cantwell v. Connecticut (1940) and Everson v. Board of Education (1947).
By 1947, therefore, the nationalization of the First Amendment was complete, with all of the rights in that amendment having been held to be applicable to the states via the due process clause.