A triglyceride is an ester made from three fatty acid molecules attached to a glyerol molecule (you get esters by reacting carboxylic acids with alcohols, and glycerol is a polyol alcohol with three hydroxyl groups). A phospholipid is very similar except one of the fatty acids is replaced by some hydrophilic molecule that's attached to the glycerol via a phosphate group.
The problem from a commercial point of view is that the fatty acids are mixed, but fish oil is only sold by EPA/DHA concentration. Nobody will buy natural fish oil because the EPA/DHA levels are too low. The solution is transesterification, where the glycerols are removed and replaced with ethanol. Ethanol is an alcohol with only a single hydroxyl group, so it can only form an ester with a single fatty acid. Now that all the fatty acids are in separate molecules, you can distill off the lower boiling ethyl esters of short chain fatty acids. EPA and DHA are long chain fatty acids, so their higher boiling ethyl esters will be left behind.
Theoretically could transesterify the ethyl esters back to triglycerides again, but when you can just sell the ethyl esters as "fish oil concentrate" there's no reason to.