How does freelance work in the UK get taxed for U.S. residents? What forms need to get filed for this? (I am a U.S. resident who wrote 2 articles for a company in the UK)

You do not need to do anything different for US tax purposes.  Unless you had to pay foreign tax on your income in which case you can claim foreign tax credit on the income to help offset double taxation.

You would enter the income in TurboTax as self-employment income.

1. Click on Federal on the left hand side of the screen
2. Click on Wages & Income at the top of your screen
3. Scroll down (you may need to click on All Income) to Self-Employment Income