If a US citizen lives somewhere else how is he supposed to pay taxes to the country he lives in? Only to the US usually? Double tax treaty? I bet many countries do not like that. Just asking out of curiosity.
Taxes paid to another country are (generally) deducted from the tax to pay to the US. There is also a rather high total income exclusion. In practice most US citizens living and earning money overseas do not pay significant tax to the US, but they must still file the paperwork (most expats who renounce their citizenship do so because of the expense/time to do so, as well as the difficulty of banking as a US citizen overseas due to the restrictions the US effectively enforces on foreign banks who have US citizens as customers).
Some countries have tax treaties. For those that don’t, you can deduct foreign tax paid from your US income taxes. In most cases, you won’t end up paying again in the US, if you already paid overseas, unless you have large amounts of income that are US-taxable and exempt in your country of residence.