The "repeated addition" idea only really works for whole numbers, though it can be stretched a little to cover negative numbers and fractions. In my mind, the easiest way to do so for fractions is to think of multiplication as "A of B". Multiplying 3 times 5 can be thought of as adding 3 sets of 5 items: 5+5+5=15. Multiplying 1/2 times 6 can be thought of as one half of 6 items, which is 3. There are a number of ways to get this result; one that arises naturally out of this formulation is that "1/2 of 6" means "one of every 2" of the 6. If we split 6 into 3 sets of 2, and take one from each, we end up with 3. This leads to the idea of multiplying by the numerator and dividing by the denominator.
This is just an informal way to approach it; I don't know how it was first thought of, and it took a long time for a reasonable notation to be arrived at and commonly adopted.
The benefit of multiplying fractions is that most measurements are not whole numbers, so we have to be able to do all the operations with them!
