Clapton is saying that we live on through our children, generation after generation. He "knew" his father (finally) when he looked in his son's eyes, and realized they were his father's eyes.
Clapton was raised by his grandparents, who he thought were his parents until he was in his teens. Growing up, he never knew his father, who only contacted him after Eric was famous. The sadness at not having a relationship is evident in this song.
In the lyric, "Just a toe rag on the run," "Toe Rag" is a slang term for a small child, street child, perhaps, in England.
Suggestion credit: Steve - Long Island, NC, for all above