As it turns out, the “or the people” part was in fact an afterthought. It was written in by hand shortly before the amendment started circulating among the legislators considering it. This was perhaps to bring it in line with the Ninth Amendment, which was more explicitly meant to prevent anyone in the future from claiming that people only had those rights enumerated explicitly in the Constitution.
As for the meaning of the Tenth Amendment itself, it is a declaration of federalism: the federal government has only those powers given to it (expressly or implicitly) by the Constitution itself.* It is also an expression of the social contract theory that was underlying their thought process. Under this tradition, a state can only have those powers given to it by those who are party to the contract. Thus any powers not given to the federal government must be retained by the states. And those that do not belong to the state (or that cannot be held by states) must thereby belong to the people.**
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* The Framers explicitly debated and rejected the notion that there are no implied powers.
** Which powers the people retain and which belong to the states is a matter of what the social contract at that level of society looks like.