I'll try to explain this as well as I can. I'm neither a native English speaker nor an environmental specialist, so bear with me.
Climate change is NOT, and never was man-made. It IS however man-influenced. When you make entire cities, you obviously influence the world around you. CO2 always was a thing, and it always caused a green-house effect. And that's a good thing. We need the earth to be somewhat warm, so that it can be habitable. The problem started when we mass-produced cars and other things with engines. Of course we need them, because most of the time, public transportation is highly inefficient crap. (Perfectly sound theory, poor execution.) Anyway, the CO2 was previously in enough quantity to keep us warm, yet little enough to let some heat escape, therefore keeping us cool. Now, it's in such excessive amounts, that it doesn't let heat escape. Remember, heat is energy, and energy doesn't get destroyed, it just changes forms and goes elsewhere, unless it doesn't have anywhere to go, of course.
As for some places actually being colder than before, you can experiment with that yourselves. Get a glass of warm water (or any drink of your choice), and put some ice cubes in it. The ice cubes will slowly melt, but as they do, the liquid will cool. However, if you leave it after they've melted, it will get warm again eventually.