With some grains and nuts it is very easy to see where the oil comes from. For example, if you squeeze a sesame seed or a sunflower seed between two sheets of paper, you can see the oil. Corn isn't quite that oily, but it does contain oil.
A kernel of corn has an outer husk surrounding a white or yellow starchy substance. At the core of the starchy substance and toward the pointy end of the kernel is the germ. The germ contains a small amount of oil. If you cut a popcorn kernel in half, you can see the husk, starch and germ. If you cut out the tiny piece of germ and squeeze the germ on a piece of paper, you will see the oil!
Corn starch comes from the starchy part of the corn. You are right that corn contains no syrup directly -- the syrup is made from the starch. In the article How Food Works, you learn about starch. Starch is a long-chain carbohydrate -- a polysaccharide. A polysaccharide is simply a collection of glucose molecules chained together. Inside the stomach, a polysaccharide is cleaved into its individual glucose molecules by enzymes so the glucose can enter the bloodstream.
To make corn syrup, enzymes are added to corn starch, and it is turned into a syrupy mixture of glucose, dextrose and maltose.
Here are some interesting links: