The usual rule, for not just the US but almost all countries, is that tax residents of a country (for some definition of tax residence) are taxed on their worldwide income, whereas tax non-residents of a country are taxed only on their income from that country. So if you have income from a different country than you reside, it is usually the case that you are taxed on that work income by your country of residence, as well as the country where the income came from. (And yes, you can have income from a different country than you reside. For example, you can commute from one country to another country to work. Or, you can have non-work income like investments in a different country than you live.)
In addition to taxing residents on their worldwide income, the US also has the rule that US citizens are also taxed on their worldwide income, even if they are not resident in the US. This unusual rule is what that statement is trying to express.
Just because you, as a US citizen, will be taxed by both the US and Vietnam on your income from Vietnam (or income from other places while residing in Vietnam), and there is no tax treaty, doesn't mean you will pay double tax. You can use the Foreign Earned Income Exclusion and/or the Foreign Tax Credit to reduce your US tax from income that is taxed by both the US and a foreign country. (You can choose one or the other or the combination of both that gives you the most savings.) In particular, the Foreign Tax Credit reduces your US taxes by the amount of foreign taxes paid on double-taxed income, or the US taxes paid on that same income, whichever is less. The result is that you pay net taxes equal to whichever is of the two taxes is higher. So you will do at least as well as paying only one country's worth of taxes.