There are various forms of felting though in all cases microscopic hairs, that exist in almost all natural fibers, are brought near each other and are linked together. Needles or agitation bring these hairs close and help them link and therefore shrink the fiber into a tight fabric.
Needle felting is when short shingles of raw wool are made into fabric using long needles that have small hooks on the shaft. These hooks catch the wool and bring the miscroscopic hairs together and compress the fiber. People make 3D figures using this technique but you can also use it to make a 2D piece. You can use the needles, wool, and your backing to needle-felt a part of your hooked mat and then hook around it. One needs stiff foam in back of the wool as you needle felt so you do not get stabbed by the needles as you jab into the wool.
Wet felting can be used to make 3D things like slippers but also 2D pieces of fabric that can be sewn to backing as a part of a mixed media piece. Wet felting is when you pull shingles of wool in various directions and pile them up. Once you have the amount you want then you wet the wool with soapy water and agitate it using your hands and maybe later using mechanical devices. You end up with fabric that can be used as is or cut into pieces.
People also felt old sweaters or knitted pieces by just washing them in hot water. Mittens are commonly made with this technique. It is the same basic process.