Why is the amount in a mole equal to the Avogadro's number?
Avogadro articulated the concept that is Avogadro's number, but didn't pick the number. From Wikipedia:


  Avogadro's law states that, "equal volumes of all gases, at the same temperature and pressure, have the same number of molecules".


https://en.wikipedia.org/wiki/Avogadro%27s_law

The Number is a logical extension of this law: So what is that number in relation to the volume and what is the mass of that amount of material?  By agreement, a mole is defined as the mass, in grams, equivalent to the atomic weight of a substance. This was laid out before Avogadro's Number was actually determined.  Then it was simply a matter of measuring the mass of a single atom to determine how many atoms were required to have a Mole of a material.  Over time, Avogadro's Number has been refined to more and more accuracy.