I think perhaps your friend misheard what the instructor said. He was probably referring to interstate transport, not "driving on the interstate."
In which case he is referring to federal law (18 USC 926A), which states
Notwithstanding any other provision of any law or any rule or regulation of a State or any political subdivision thereof, any person who is not otherwise prohibited by this chapter from transporting, shipping, or receiving a firearm shall be entitled to transport a firearm for any lawful purpose from any place where he may lawfully possess and carry such firearm to any other place where he may lawfully possess and carry such firearm if, during such transportation the firearm is unloaded, and neither the firearm nor any ammunition being transported is readily accessible or is directly accessible from the passenger compartment of such transporting vehicle: Provided, That in the case of a vehicle without a compartment separate from the driver’s compartment the firearm or ammunition shall be contained in a locked container other than the glove compartment or console.
This federal law supercedes state law, such that you can always transport a firearm, unloaded and locked in the trunk, between two states where you are legally allowed to possess the firearm - regardless of the laws of the state through which you are traveling.
An example of this would be a person driving from Pennsylvania to New Hampshire, through New York. In New York, it is illegal for anyone to possess a handgun without a NY license, period. However, the person can transport the gun through New York if it is unloaded and locked in the trunk of the vehicle.
Of course, if a state allows you to carry a firearm in the passenger compartment, you may do so. However, laws vary from state to state, and the "easy answer" is that you can always carry the gun unloaded and locked in the trunk.
The "real answer", it's that you need to check the laws for every place you drive through, but discussing the local laws everywhere in the country is far beyond the scope of a CCW class.
One important thing to note however - Washington, DC is not a state
and this does not apply
there. Drive around it.