Why did thomas paine write the pamphlet common sense?

There are several different reasons, but in the end, they all equal to one thing. That was to fight for independence from Britain.
Thomas Paine wrote "Common Sense" to rally the cause of independence from Great Britain to people in the Thirteen Colonies. Paine wrote the pamphlet to change the minds of people who wanted to peacefully settle their differences with the British government to fight for independence instead. 

He wrote it to help free america from King George. Hope this helps! (tbh the first thing i thought of was the Angelica) (from Hamilton)