can someone explain the principle behind this multiplication method?

It's using FOIL. See: http://en.wikipedia.org/wiki/FOIL_rule

The problem 21 × 13 can be seen as:

(2×10 + 1) (1×10 + 3)

Watch:

100s: 2×1 = 2 (F)

10s: 1×1 + 3×2 = 7 (OI)

1s: 3×1 = 3 (L)

FOIL. It's that easy. The number of lines that intersect in a particular column make up the F, OI, and L parts. This is because if you have n parallel lines and m parallel lines intersecting like that, you get n× m intersections.

200 + 70 + 3 = 273

For larger ones, it works the same way, but using FOIL for larger expressions.

essentially if you draw m lines running northwest to southeast and n perpendicular lines (NE to SW) then they will create mn intersections. the square array uses this technique to multiply every digit of one number by every digit of the other.

then he knows that the ones digit of the product is the last digit in the product of the ones digits. the tens digit of the product is the tens digit of the product of the ones digit + the sum of ones digits of the droducts of the tens digit of one number and the oens digit of the other. continue in this manner, adding the carry from the previous step to those digit products that contribute to each digit of ther asnwer.