The United States taxes its citizens on worldwide income no matter where they live — almost uniquely among countries. The Foreign Earned Income Exclusion, claimed via IRS Form 2555, is the main relief valve for Americans living and working abroad: it lets them exclude a set dollar amount of foreign wages or self-employment income from US income tax each year.
To qualify, you need a tax home in a foreign country and need to pass either the Bona Fide Residence Test (established residency abroad for a full tax year) or the Physical Presence Test (at least 330 full days outside the US in any 12-month period). If you qualify for only part of the year, the exclusion limit is prorated.
The FEIE only reduces income tax — it does not reduce US self-employment tax, which self-employed expats still owe in full unless a specific totalization agreement applies. Expats in high-tax countries sometimes do better with the Foreign Tax Credit instead of, or alongside, the FEIE.