Cleansing oils work because they're an ingenious combination of water solubility and oil. All oils have similar molecular bonds (specifically covalent ones), and as a result, are capable of breaking apart similar molecules. So when you rub on an oil cleanser, it dissolves even tough makeup and breaks up sebum and grime on the surface of your skin. Then, because the oil has been made water soluble, the entire formulation washes away completely. Your skin gets clean without getting stripped of its natural moisture the way it does with some traditional cleansers.
I especially love cleansing because I have dry, sensitive skin that's also prone to acne (fun!), so oil cleansers are a great way to get clean without getting dried out and irritated. Right now, I'm using DHC Deep Cleansing Oil ($12.50), but Shu Uemura Cleansing Beauty Oil ($34) is also great. And if you love the results of cleansing oil and want to go even more natural, think about trying the Oil Cleansing Method.