While many people argue that the right to bear arms is a "constitutional right," this is not accurate. There really is no such thing as a "constitutional right," only unalienable rights that are further protected by the constitution.
When the founding fathers wrote the constitution they specifically granted rights to the Federal government. These rights are spelled out quite plainly, including the ability to collect taxes, establish Post Offices, maintain a military, to coin money, etc. Any rights not explicitly given to the Federal government are reserved to the states and to the people.
The Bill of Rights was a hotly debated addition to the Constitution. One group thought that the addition of the Bill of Rights would later be interpreted to mean that those rights were somehow granted by the government. They felt that if they added any rights to the Constitution the government would assume that those were the only rights granted to the citizens. The other group felt that if no rights were listed that the government would not recognize any rights. The ninth and tenth amendments (ninth: "The enumeration in the Constitution, of certain rights, shall not be construed to deny or disparage others retained by the people" and tenth: "The powers not delegated to the United States by the Constitution, nor prohibited by it to the states, are reserved to the States respectively, or to the people") were put into the Bill of Rights as a sort of compromise.
These two amendments are, to some extent, more important to the people's right to bear arms than the second amendment. No where in the Constitution does it give the Federal government the right to control or restrict arms. Because the Constitution does not grant this power to the government, it is automatically granted to the states and the people. The second amendment restricts the states from passing laws that infringe upon the rights of the people, so the right is retained by the people.
A person has an inherent, unalienable, and natural right to life. This right includes the right to defend themselves from unprovoked aggression. Since an aggressor is likely to have a firearm (or other weapon), it is reasonable to grant a regular citizen to have similar weaponry in order to defend themselves (after all, if a person is being shot at, there isn't much use in trying to fight back with a butter knife). The founding fathers of this country understood that people need to be equipped to defend themselves from aggressors, whether it is another citizen, a foreign army, or their own government. This is the reason that the right was explicitly spelled out in the Bill of Rights. But, the right would exist whether it was included in the Bill of Rights or not, specifically because no where in the Constitution is the government granted the right to restrict the people from bearing arms.