This notion of rights that the founding fathers had was influenced by Enlightenment philosophers like Locke, Voltaire, Hume, Rousseau and so on. (see The Philosophy Behind the American Revolution)
The concept was that rights cannot be granted, nor taken away by a document. In fact, many of the founding fathers were troubled by the idea of a "bill of rights" and it wasn't included in the Constitution. It was controversial at the time and there are some names that you might expect to find as signatories to the constitution who refused to sign because of this. The debate was between the "Federalists" and "Antifederalists".
So when the Bill of Rights was written and submitted as a series of amendments, the language is such that it recognises the existence of a right and gives it a legal footing. It doesn't create the right. And it explicitly states that there are other rights.
XI. The enumeration in the Constitution, of certain rights, shall not be construed to deny or disparage others retained by the people.
So, for example, the second amendment states that the right of the people to keep and bear Arms, shall not be infringed. The existence of the right is independent of its enumeration, it is assumed to exist. The amendment gives legal power to courts to strike down legislation that infringes on this right.
The Declaration of Independence follows the ideas of Locke etc. and states in the broadest terms that "people have rights". The Constitution recognises some specific rights and gives the courts the power to review legislation that infringes these rights.
The enumerated rights are among the "certain inalienable rights" of the DoI, but there may be others which are unenumerated. In particular, the "pursuit of happiness" is not enumerated in the Constitution.
The question of unenumerated rights has never been properly settled. The supreme court has not used the ninth amendment to give a judicial review to legislation, preferring to use the first 8 clauses of the Bill of Rights or other parts of the Constitution, such as that requiring "due process". There are at least two major interpretations: One, is that as this is an explicit statement that there are unenumerated rights, it is inappropriate for judges to enumerate them. This is the "inkblot" theory, If a clause of the contract is hidden by an inkblot, judges should not speculate on what was hidden. The other interpretation is that this is an invitation to give the unenumerated rights the same legal protection as the enumerated ones - and in doing so, significantly restrict the application by Congress of the "necessary and proper clause" - See Interpretation of the Ninth