Oils, whether they be vegetable and plant oils in a jar, or oils that make up the cell membranes of an organism are prone to oxidation and degradation. This oxidation can be stopped by antioxidants, particularly vitamin E. 
Oils are mainly triglycerides which are composed of fatty acids. These fatty acids are carbon chains that consist of either double bonds or single bonds between the carbons.  Carbon chains containing single bonds are known as saturated fatty acids while those with one or more double bonds are referred to as unsaturated. It is the unsaturated fatty acids that are more prone to oxidation and the risk of oxidation increases with the number of double bonds.

This oxidation is initiated by light, heat, metals or oxygen, so keeping oils away from these helps to slow oxidation, but it is inevitable. A free radical contains an unpaired electron which makes it highly reactive and unstable. It is looking for an electron on another molecule to steal, thus setting up a chain reaction of forming free radicals. When these oxidative products are formed from oils the breakdown products are off flavor compounds such as ketones, aldehydes, alcohols and more. They tend to have a characteristic smell known as ‘rancid’.
Your body is constantly making free radicals, and at the same time making antioxidants to stop them from causing damage. Hopefully, a nice balance exists, but not always. The oil in a bottle on your shelf or your newly made skin care product has no way to make antioxidants like your body does so it is helpful to add an antioxidant to prevent oxidation.
Vitamin E is an important oil soluble antioxidant to protect polyunsaturated fatty acids and other cell membrane components from oxidation. Vitamin E reacts with a lipid radical faster than that lipid radical can react with other lipids and so suppresses the propagation of oxidation.  It does this by transferring a hydrogen atom to a lipid free radical. Of course now the vitamin E molecule becomes a free radical but it can react with another vitamin E free radical to form a non-radical product. This terminates the chain reaction of oxidation.
Vitamin E is a family of eight different molecules; 4 tocopherols and 4 tocotrienols. Vitamin E is high in wheat germ oil, canola oil, and almond oil, but commercially, vitamin E is typically extracted from soybean oil. Buy your vitamin E from a skin care ingredient supplier, do not use capsules from the store as this is not pure vitamin E.

Bottom line is that your skin care products should contain an antioxidant such as vitamin E to prevent the breakdown of oils in your product. Many crafters are being taught that this is similar to a preservative that will prevent the growth of bacteria or fungus in a product where it is not at all similar. If you have a product that contains any form of water, you need to have a preservative to prevent growth of microbes. Vitamin E will not prevent growth of microbes, but vitamin E should be in your products that contain oil to prevent the breakdown of these oils.
To learn more about preservatives (anti-microbials) see this post on What is the Best Preservative?