When someone is being patronizing, they are finding a rude way of talking to someone. They are intentionally using certain words and tone to make the person they're talking to feel a little stupid. Therefore, the recipient of the treatment would say, "Don't patronize me," to let them know they don't appreciate how they're being spoken to.
An example of this is a person telling another that they might have to walk to the store because they can't give them a ride. They might say something patronizing such as: "Walking. Do you know what I mean? You'll have to put on your shoes...put one foot in front of the other...take about 300 steps up the street...and get to the store yourself?" This could leave the other person feeling patronized, and they could respond with, "Don't patronize me."
Here's a personal example that I'll never forget from a patronizing border security guard. I was asked "How many bags are you bringing over the border?" Bags. Well, I had two pieces of travel luggage... and my purse. I wanted to be concise. So I explained exactly that. She asked, "So how many bags?" I said, "Two. But I also have my purse." She finally said, "Do you know how to count? What's two plus one?" That last line about asking a grown woman about counting and getting her to answer a Grade 1 math problem is patronizing. It's to make someone feel completely foolish for not knowing something very simple -- when they very likely already do know. (I very much wanted to say "don't patronize me" but no border guard wants back-talk from anyone, especially so if they deserve it!
Usually patronizing remarks come after some poor explaining on the patronizing person's part.