The insulin protein promotes glucose (not fats) uptake into the muscles (~75%), liver (~15%), and other tissue (~10%) to aggressively lower blood sugar levels back to a total circulating amount of ~5g.
In the obese, glucose is preferentially directed to adipose tissue for storage as fat. This is due to both insulin resistance, and to the limited storage capacity in muscles and the liver for glucose in the form of glycogen. I was writing colloquially and intentionally left out details like lipogenesis, which in retrospect probably wasn't a great idea.