Why does 1 amu = 1 g/mol?

It was made that way

##M_(c12) = 12 (AMU)/(ATOM) = 12 G/(MOL)##

In other words, the ratio of amu/atom is the same as the ratio of g/mol. The definitions of amu and moles were intentionally chosen to make that happen (I'm surprised your teacher didn't explain this, actually). This allows us to easily relate masses at the atomic scale to masses at the macroscopic scale.

To check this, look at the mass of an amu when converted to grams:

##1 (AMU) = 1.6605⋅10^(−24)G##

Now divide one gram by one mole:

##(1 G)/(1 MOL) =( 1 G)/(6.022⋅10^(23) ATOMS) = 1.6605⋅10^(−24)G##

Same number!