It helps if we think of multiplication as finding the area in square units of a rectangle - i.e. multiplying the number of squares in one row (the length) by the number of rows (width).

Then we need to define a unit square to be a square with length and width each = 1 unit in length.

Say we wanted to evaluate , we need to evaluate the area of a square that has a length unit in length, and width unit in length.

How much of the unit square ? Why, of it. Therefore .

Now what if we tried something like ? We would need to evaluate the area of a rectangle that has a length of of a unit, and a width of of a unit.

How much of the unit square is covered by this rectangle? Why, . Therefore .

I'm sure you're seeing that it looks like you can multiply tops and multiply bottoms. Why is this?

It's because in order to multiply the fractions, you need to split the "length" of your unit square into as many pieces as defined by the denominator of the first fraction, and you need to split the "width" of your unit square into as many pieces as defined by the denominator of the second fraction. That means the unit square is divided into as many pieces as the product of the denominators, and so this becomes your new denominator.

Then you need to count as many of these pieces along the "length" of your unit square as in the numerator of the first fraction, and you need to count as many of these pieces along the "width" of your unit square as in the numerator of the second fraction. So the number of pieces you will be counting in total is the same as the product of the numerators, and so this becomes your new numerator.

Therefore, to multiply fractions, you can multiply the numerators and multiply the denominators.