"At" wouldn't be correct here. When you "look at something", you are "watching " it. You can't watch your age.
Here it means you are starting to look like you should be at your age. If you are 50, you may have a few lines on your face, some streaks of white/gray in your hair.
If you're 80, your hair's bound to have turned white/gray, you may be stooped, etc etc...