In theory, these rights are granted to US citizens by the Constitution and its amendments. As to whether or not US citizens actually have these rights, Constitutional scholars can and will argue back and forth on that without making a real decision, partly due to the non-specific nature of the rights described in your quote.
It's important to note that the quote you provided is actually from the US Declaration of Independence, which declared that the 13 colonies were no longer going to be British colonies, but rather their own separate nation (keep in mind that from a British perspective, this was treason).
The actual rights that US citizens possess are granted primarily by the US Constitution and its amendments (especially the first ten, also known as the Bill of Rights). This forms the basic foundation for our governmental system, on which all other parts of it are built. While many people feel that the wording of the Declaration of Independence should be used to help interpret the Constitution, there's nothing in the Constitution that formally states that. Historically, it's also important to recognize that the current Constitution is not the original one - the first document formalizing the federal government of the US was the Articles of Confederation, which was the law of the land for over a decade before being superseded.