Why does the First Amendment not override the Copyright Clause, making copyright an unconstitutional violation of freedom of speech and press?
The Supreme Court generally treats the Copyright Clause and the First Amendment as complementary, not conflicting:
The Copyright Clause and First Amendment were adopted close in time. This proximity indicates that, in the Framers' view, copyright's limited monopolies are compatible with free speech principles. Indeed, copyright's purpose is to promote the creation and publication of free expression. As Harper Row observed: "[T]he Framers intended copyright itself to be the engine of free expression. By establishing a marketable right to the use of one's expression, copyright supplies the economic incentive to create and disseminate ideas."
Although the Court didn't remark on it, this treatment is also consistent with the canon called generalia specialibus non derogant, "the ancient interpretive principle that the specific governs the general." Nitro-Lift Techs., L.L.C. v. Howard, 568 U.S. 17, 21 (2012).
Here, the First Amendment supplies a rule that Congress may not abridge the freedom of press generally, but the Copyright Clause supplies a rule allowing Congress to abridge the right to use specific writings. The Copyright Clause is more specific, so it creates an exception to the First Amendment's general rule.
In addition to the answers by bdb484 and AnoE (both of which I agree with), the Constitution is generally interpreted as a consistent whole, which does not contradict itself. The amendments are generally treated as adding to the Constitution, but not as "overriding" parts of it, except where they clearly or explicitly modify an existing provision. So the direct election of Senators changed the previous provision for their appointment by the state legislators. The separate election of the Vice-President replaced the earlier provision for Presidential elections. The amendment allowing an Income tax modified the previous (never used) provision for apportionment of direct taxes. But aside from these and a few similar cases where an amendment clearly or explicitly overrides a prior provision. amendments generally add to the constitutional fabric, but do not override existing provisions.
To answer the question...
The government can pass copyright laws to restrict certain aspects of reproducing works with no issue regarding freedom of speech. In this particular case, keep in mind that copyright specifically protects works (in the technical sense), i.e., the concrete, physical formulation of text, images etc. - it says nothing about ideas, facts, opinions and such. Famously, algorithms, cooking recipes etc. cannot be copyrighted precisely because copyright does not apply to the factual content (only the representation). Freedom of Speech protects speech, opinions and such, so in a sense the exact opposite. As an example, copyright does not keep you from reading a book, and repeating the content in your own words. No conflict there.
Copyright would only protect speech in a very specific sense - i.e. if you have a recording of a particular speech or piece of literature etc.; that particular audio recording including the concrete choice of words (the "work") would be subject to Copyright. Everybody else would be totally free to say anything similar to that in their own words.
The same applies if you interpret "speech" independent of audio - i.e. you can absolutely take a book, reformulate every sentence in your own words, and publish it to your heart's content. As it is a completely different work, copyright would not be involved in that.
Speaking more generally...
This question shows a very common misunderstanding of what "free speech" in general, or the First Amendment of the US constitution is.
To quote the first sentence on the amendment in Wikipedia:
The First Amendment (Amendment I) to the United States Constitution prevents the government from making laws that regulate an establishment of religion, or that prohibit the free exercise of religion, or abridge the freedom of speech, the freedom of the press, the freedom of assembly, or the right to petition the government for redress of grievances. It was adopted on December 15, 1791, as one of the ten amendments that constitute the Bill of Rights.
The focus of the 1st amendment is to limit the right of the government to pass certain laws. It is not a free-for-all pass for everybody to say everything.
In general terms, Freedom of Speech does not mean that everybody can just say anything at all at any point of time without repercussions.
In other words, to quote the more general entry on Freedom of Speech, Freedom of Speech is not an "absolute" right but has to be seen in the context of what is being said by whom:
Freedom of speech and expression, therefore, may not be recognized as being absolute, and common limitations or boundaries to freedom of speech relate to libel, slander, obscenity, pornography, sedition, incitement, fighting words, classified information, copyright violation, trade secrets, food labeling, non-disclosure agreements, the right to privacy, dignity, the right to be forgotten, public security, and perjury.
To recap: Freedom of Speech in the context of constitutions means that the government cannot restrict their citizens in very specific ways that would - basically - prevent citizens to express critique of the government, to voice their opinions, to assemble, and so on. One of the most important words in this is "government". It can be quite legal for other entities (companies, clubs etc.) or rulings (trade secrets etc.) to restrict said freedoms; sometimes even for the government (in the case of copyright, or classified information etc.).
Examples where no such freedom exists:
- Having active moderation in online forums etc. does not fall under the Freedom of Speech. For example, a forum about CPUs from company 1 could decide that they do not allow even mentioning CPUs from company 2 (for whatever reasons). A forum for runners could decide that asking medical or highly individual questions (with no benefit to anybody else) would be forbidden. Completely legal, both practically as well as philosophically; not related to Freedom of Speech whatsoever.
- Companies can regulate their employees to keep them from talking about internal topics with no limitations whatsoever (and often do, even for external entities like reporters, or subcontractors, in the form of NDAs).
- While lying is generally not considered a crime, in many countries you cannot write certain lies on the packaging of food (i.e. lies about the weight/volume of the contents, or the exact ingredients - they are still free to lie in other aspects, e.g. calling a food "healthy" which is clearly detrimental to many people, i.e. primarily sugar-water-based drinks).