Literally, when you go skating on thin ice you are taking a big risk. You need to be extremely careful. If you make the slightest mistake, or fail to judge the thickness of the ice perfectly, you will break through the ice and fall in. (You will quite possibly die from hypothermia).
"I feel like I am skating on thin ice with her" means "I am afraid if I say something wrong--I'm afraid if I don't do everything exactly right--she will be angry and reject me."
Two other idioms with similar meanings are "I feel like I'm walking on eggs" and "I feel like I'm tiptoeing through a minefield."