Sort each expression by what it counts.

Counts subsets: Two to the power of the size of A

Counts ordered pairs: The size of A times the size of B

Counts functions: Each input independently picks one of the outputs

Subsets go as $2^{|A|}$ because each element is in or out; pairs as $|A|\cdot|B|$; functions as $|B|^{|A|}$ because each input independently picks an output. All three are the same independence argument applied differently.