He was singing while caring for the sheep.

caring for the sheep,看护绵羊，这句话的意思是：他在看护绵羊的时候唱着歌

He was singing while (he was) caring for the sheep.
意思就是他边唱歌边看护着绵羊.

他一边放羊一边唱歌
改成从句是He was singing while he was caring for the sheep

while caring for the sheep是本句话的状语，完整的应该是while he was caring for the sheep，因为前面已经知道是he，所以后边状语省略它，这也是状语运用的一条规则，理解为：他边放羊，边唱歌。

caring是care的变形，句子可翻译为“他一边唱歌一边放羊”，可换成从句为“He was singing while he was caring for the sheep.”

The mother is listening to the radio while caring for her baby.