So, when you get a burn, your skin is trying to protect itself from further damage. It's like, your skin is saying, "Hey, I've got this, just give me a minute." And, in that minute, your skin starts to heal itself by increasing blood flow to the affected area. But, if you put ice on it, you're basically telling your skin to stop doing its thing.
Ice causes the blood vessels to constrict, which reduces blood flow to the area. And, guess what? That's the opposite of what your skin needs to heal. It's like, you're sabotaging your skin's natural healing process. Ouch, right?