As mentioned by all the other answers, the head is what's being referenced in the size (1/2", 3/8", ...) and the bottom is intended to be inserted into the chuck of a drill/driver or even a manual handle.
The reason why it's a common size is for maximum flexibility of usage but in many cases not at maximum torque. You see this often in YouTube videos where a long wrench is needed to loosen a bolt, but then a small driver/drill is used to completely remove the bolt.
The pictured attachment is a good example of something to help insert/remove something that is "finger tight" a larger fastener but would probably not be reasonable to tighten to higher torque specifications required by most larger sizes. I will also work fine for smaller sockets and fasteners where the driver will stall before the fastener or socket breaks.
The size of impact driver you've purchased will be limited to lower torque values (e.g. less than 100 ft/lbs, 100 N.m) so this adapter will work for most of the sockets you will be able to purchase for that size of driver. Most impact sockets are 1/2" or larger because those fasteners usually require significant torque (i.e. a bar the length of your forearm or entire arm) and will break if too small.
Almost forgot to mention: Don't use a larger driver on something too small unless you enjoy breaking fasteners/nuts and having to remove them. Unless you know to use higher torque, it's cheaper to use less and let the tool/fastener tell you that it isn't enough, than to mash it and snap things off. Fixing a snapped bolt/screw or stripped head is much more annoying than one that's half in and the driver stopped because you have the setting too low.