Why can a airplane designed to fly extremely fast have smaller wings than an airplane designed to go much slower?

Because a smaller plane has considerably less drag than a massive plane. Because of their small stature, they have less air they have to force their way through.

But then again, most big planes have propeller systems instead of afterburners. If you take a look at the Concorde or the Tu-160 Blackjack etc.. They have powerful engines/a body specially designed to easily cut through the wind.

This is why the X-43 is the fastest non-spacecraft in the world, travelling Mach 9.6 as a top speed, yet about a head smaller than your average full-grown man. (However, it is launched from a missile launched from a B-52.) Same thing with missiles.

So basically, big planes have more air they need to force through while smaller planes have less. It's all about the weight of the plane, the engines and the shape.