Why does a voltage buffer has an high input impedance?
You probably assume, like many beginners, that the only "useful" amplification is voltage amplification, since that is what is taught first (usually).

The CC configuration, more commonly known as emitter follower is an example of an amplifier that doesn't provide voltage gain, but current gain instead, hence also power gain. It is also known as (voltage) buffer (amplifier).

BTW, that's the same principle used for buffer gates in digital logic: they are useless from a strict logic POV, since the logic level at their input is the same as the one at the output. Their role is purely electric: they provide current gain (and delay).

Keep in mind that electronics is a complex subject: it is nearly impossible to reduce all its complexity to a bunch of basic ideal circuits. Of course you cannot dump tons of stuff on a beginner, so they teach the basics, and this tends to hide much of that complexity ("amplification is only voltage amplification" is one of that "lie to children things").