The Bill of Rights modified the original Constitution with respect to religion through the First Amendment. That amendment guaranteed freedom of religion in two ways. It guaranteed the right of free exercise of religion and it guaranteed that the government would not establish a religion. Neither of these things was explicitly stated in the Constitution as it was originally written.
In the original Constitution, religion is only clearly mentioned once. This is in Article VI where it says that there can be no religious test for holding public office in the United States. In other words, the original Constitution explicitly stated that people of any religion (or, presumably, no religion) had the right to hold office in the US. This hints at both the Free Exercise clause and the Establishment clause, but it does not explicitly do what either of those clauses of the First Amendment does.
In the Free Exercise clause, the First Amendment explicitly states that Congress cannot make any laws that infringe on people’s rights to practice their religion. In the Establishment Clause, it explicitly states that the federal government cannot establish a religion (this means to set a given religion as the official religion of the country). By guaranteeing these rights explicitly, the Bill of Rights modifies the original Constitution on the issue of religion.