In addition to the above comments, I like to think the Bill of Rights is modeled on human rights, as opposed to merely American ones. By at least attempting to adhere to these rights for all peoples instead of just citizens, we ensure that we remain something closer to a democracy, a nation that practices and protects actual freedoms rather than symbolic or philosophical ones. I think this is a much more successful approach to exporting democracy to other nations than doing so by force of arms.
As other editors have suggested, the Bill of Rights is actually one way in which the United States of America tries to ensure that it is morally principled and just towards all people, rather than differentiating between US citizens and other people, such as refugees and immigrants. The US has modelled itself as a haven offering sanctuary to all who want shelter from the metaphorical storms of life, so it would be dangerous to suggest that there are two levels of treatment for people in the US.
People who are in the United States should absolutely have the same basic rights as those guaranteed to American citizens by the Constitution.
These rights, such as freedom of speech and religion and the right to be free from unreasonable searches and seizures are given to us not simply because we are American citizens. They are given to us because we Americans believe that our government should not violate these rights in general. We do not want a government that searches people's homes without warrants. We do not want a government that suppresses free speech. If we allow our government to do these things to non-citizens, we are encouraging it to form habits that could ultimately be turned on us as citizens.
The Bill of Rights exists at least in part as a way to prevent our government from becoming tyrannical. We do not want our government to get used to exercising the powers of a tyranny and so we do not want it to be tyrannical towards anyone who is in our country.