What's actually happening when you sharpen a knife?
When you sharpen a knife you are removing steel from the edge in order to turn it from a rounded, dull edge into a sharp one. When talking about sharpening angles a lot of times it gets confusing trying to figure out how the angle is being measured. Normally the angle in question, for instance 15°, is only half of the total angle and is referred to as "degrees per side" (in knife sharpening the total angle is called the "inclusive" angle). So if a knife has a 15° per side, 30° inclusive edge angle then what is really meant is that if you imagine a line running through the middle of the knife from spine to edge, the angle from that line to the outside is 15° while the angle from one edge to the other is 30°.
What is re-profiling?
Re-profiling is a process in which the edge angle is made more obtuse or acute than the current angle. Most of the time when we re-profile a blade we are making it thinner because most knives, kitchen knives especially, come from the factory with a bevel angle that is more obtuse than ideal. Knife manufacturers do this to ensure that people who use their knives incorrectly don't accidentally damage them and blame the knife for being fragile. With proper usage, knives can benefit from a thinner angle because there will be less drag when cutting which improves cutting performance.
The "Burr"
A burr is a thin, wire edge of metal that forms on the opposite side of the edge that you are sharpening. In this example, the knife sharpener has been removing metal from the red side of the edge which forms a burr on the opposite side. The formation of a burr can be felt by an experienced knife sharpener and lets them know that they are removing metal all the way down to the edge and are therefore making the cutting surface sharp. While a burr is a good indication of progress it still must be removed in order for the knife to be left with a clean edge. This is done by flipping the knife and sharpening the other side. Successive strokes on each side with lighter and lighter pressure is the key to removing a burr and doing this with each grit will result in a well-formed, sharp, and even edge.