Line Multiplication is a way to visualize multiplication that relies on the distributivity of multiplication. To perform the multiplication, the steps are:
1.

Draw separate sets of parallel lines to represent each digit of number A. For example, for the number 12 draw one vertical line for the first digit 1, and then draw 2 lines for the second digit 2. Make sure to separate the lines for the first and the second digit.

2.

Draw separate sets of parallel lines that are perpendicular to the previous ones that represent the digits of number B.

3.

Draw points where one line intersects with another.

4.

Notice the upwards diagonal blocks that form by grouping the clusters of points of intersections from bottomleft to topright. Count the total number of points in each of these diagonal blocks.

5.

Starting with the bottom rightmost diagonal block (this should just contain one cluster of points), write the onesdigit of the number of points in that diagonal, this corresponds to the onesdigit of the answer. Add the remaining (if there is any) to the next diagonal's total.

6.

Repeat this with the next bottom rightmost diagonal block, this will correspond to the tens digit of the answer. Keep doing this for the rest of the diagonals. If there's anything to carry over from the last diagonal, write the whole number instead of the just the ones digit.

7.

After doing this for every diagonal block, you will end up with the answer to A times B.

For example, if there are 3, 8, 14, 8, 3 points in five different diagonals correspondingly. We count from the right, 3>8>4(carrying 1 to the next value)>9(8+1)>3, so the final number from left to right is 39483.
The theory behind the line multiplication is the distributivity of multiplication. Notice that each number represents a digit and each intersection represents the product of two digits, by adding up all the products diagonally, one is essentially summing up the products of the same power of 10.
123 * 321 = (1*100+2*10+3*1) * (3*100+2*10+1*1) = (1*3*100*100) + (1*2*100*10+2*3*100*10) + ... + (3*1*1*1) = 39483
