The key points: high temperature oxidation protection and the improvement of low temperature flow.
To break it down: motor oil is comprised of two main components: base oil (approximately 80% of the formulation), and additives (the balance, approximately 20%). Base oils in synthetics are highly refined. In other words, they are engineered to have a specific set of properties that determine viscosity and other physical characteristics.
To begin with, our synthetic base stocks are more resistant to degradation at high temperatures -- namely oxidation, which leads to unwanted thickening. Why is that unwanted? As oil viscosity increases, it moves less freely and works less efficiently. Since synthetics are more resistant to oxidation, they help keep the oil at the desired viscosity. Additives like antioxidants enhance this protection. (An example: Zinc Dialkyl Dithiophosphates or ZDDP, is an antioxidant. It also has other properties such as antiwear, preventing metal-to-metal contact between engine parts by forming a protective film.)
On the other end of the spectrum, synthetics offer better protection for starting your vehicle in cold temperatures. Cold temperatures can cause conventional motor oils to form waxy deposits which can result in gelling. Oil with significantly increased viscosity doesn't flow as well and can starve the engine of needed lubrication. Synthetic base stocks don't contain the same waxy materials as conventional base oils, reducing the risk of increased low temperature viscosity and reduced protection.