"My, my!" means something like "Wow. I'm impressed."
However, English speakers rarely use "My, my!" when they're honestly impressed. More often, they say "My, my!" to tease someone. For example, if someone in your family usually wakes up late, you can say this when they wake up earlier:
My, my! You're up early.
Although we use "My, my!" to tease people, it doesn't seem mean or offensive. It's fun and playful.