When the so-called Founding Fathers of the US wrote the Constitution, they were used to living with political and religious oppression. They took care not to repeat the same mistakes they witnessed overseas.
The right to free speech is granted in the Bill of Rights. Flag burning does not physically harm anyone or deny their right to happiness. It is also not a form of slander, and is therefore a Constitutionally-protected right.
A similar issue is the legal status of burning a cross, a symbol of both religious and racial intolerance. As expected, it is illegal to do so on property belonging to someone else, such as the infamous 'raids' of the vile Ku Klux Klan.