On my days off, I start to become active at night, when my body has been well rested.
ご質問ありがとうございます。
"On my days off"=「私の休みの日には」
"I start to become active at night,"=「私は夜に活動的になり始める」
"when my body has been well rested."=「私の体が十分に休めたときに。」
☆又は、"when I'm sufficiently rested" (sufficiently=十分に)という言い方もできます。
ご参考に。