Amankwaah, 26, played just four times for the Swans last season as the Welsh side claimed the League One title, and is looking forward to a more prominent role at the County Ground.
"I hope I can find my best form again at Swindon. They are looking to build something special there and hopefully I can play a big part," he said.
"My first team chances were limited last season and I didn't want the same thing to happen again next season.
"I could have sat in the stand and just picked up my wages, but I want to play football."
"Every player gets a blip in their career and I hope I can find my best form again at Swindon."