What's that scum at the sides of my home-made burger when I cook it?

This scum is made from proteins. Meat contains muscle fibers (the proteins actin and myosin) as well as some loose proteins swimming in the fluids within the meat (the cell plasma). When you cook meat, the protein-rich fluids are expelled (that's why overcooking makes meat dry). Under hot temperature, the proteins in the fluid coagulate, making it firm. It is especially noticeable in your burger, because there is more liquid flowing out quickly from your cut-up meat, but it also happens with whole pieces of meat, albeit more slowly. It is also very noticeable when cooking stock, because the stock is cooked for a long time and the liquid has time to come out from the meat.

The coagulated liquid can form a single piece (as it does with steak on a slightly oiled pan), but when it flows into water or oil, it mixes with it without dissolving, creating small loose particles. They float to the top, creating the frothy scum you describe.

The process is perfectly natural, you're not doing anything wrong. You can in fact eat the scum without any ill effects, but the taste isn't that great. So it is better to fish it out from the oil (or water when making stock). In dry cooking (aforementioned steak) it is exactly this stuff (after getting nicely browned on the hot pan) which makes gravy taste so well, together with the dripped fat.