Product rules in Combinatorics: Why do we multiply and not add or divide?

For each letter of the three letter word, there are two choices, A or B. Hence there are $2^3=2\cdot 2\cdot 2$ choices.

First letter may be A or B, so we have 2 options: For each of those 2 options (1st letter A, 1st letter B) we have 2 options for the second letter, namely A or B:

So we now have 2 * 2 = 4 options for the first 2 letters, namely AA, AB, BA, BB:

And for each of those 2 * 2 (i. e. 4) options for the first 2 letters (AA, AB, BA, BB) we have 2 options for the third letter, namely A or B:

So we finally have 2 * 2 * 2 options: AAA, AAB, ABA, ABB, BAA, BAB, BBA, BBB:

For the first 2 letters we obtain a square 2 x 2, so there are 4 options:

     A    B   
   ┼────┼────┤
 A │ AB │ BB │
   ┼────┼────┤
 B │ BA │ BB │
   ┴────┴────┘

Now we can append either A as the 3rd letter:

      A      B   
   ┼─────┼─────┤
 A │ ABA │ BBA │
   ┼─────┼─────┤
 B │ BAA │ BBA │
   ┴─────┴─────┘

or B as the 3rd letter:

      A      B   
   ┼─────┼─────┤
 A │ ABB │ BBB │
   ┼─────┼─────┤
 B │ BAB │ BBB │
   ┴─────┴─────┘

You may imagine those two 3-letters squares as two layers of the resulting cube, volume of it will be 2.2.2 = 8.