It is best illustrated by an example.  To multiply 53 by 67, form two columns. At the head of one, say, the left-hand column, put 53, at the head of the other, 67.  Successively divide the left-hand column by 2 (until a quotient of 1 is obtained) and multiply the right-hand column by 2.  When an odd number is divided by 2, discard the remainder. 
 

Cross out the rows which have an even number in the left-hand column. 
 

Add up the remaining numbers in the right-hand column to obtain the desired product. 

This result may be verified by traditional multiplication. 

The relation of this method of multiplication to the binary system is not too difficult to discover.  First, in the left-hand column, in successively dividing by 2, we employed the same procedure used to determine the binary representation of a decimal number. 
 

Thus                           53 _decimal   =   110101 _binary     or     32 + 16 + 4 + 1 

In the right-hand column, in successively multiplying by 2, we obtained binary multiples of 67.
 

Since these are the binary multiples of 67 that we must add to obtain the product of 53 and 67, we should select and add only those multiples in the right-hand column that contribute a "1" to the binary representation of 53, that is, opposite odd numbers.

Incidentally, real Russian peasants may have tracked their doublings with bowls of pebbles, instead of columns of numbers. They probably weren't interested in problems as large as our example, though 3551 pebbles would be hard to work with!