Meaning that, if anyone who is a US citizen, this person will be treated as such in the US. That means this person does not enjoy diplomatic protection of any kind in the US. On the other hand, the US government does not care whether you use other countries' passports outside of the US. There is no requirement for you to destroy your other passport (except some special cases, such as you have DoD security clearance, you than must return/destroy other passports you process)