This thread contains two explanations:

By user typographicalerror:

Every prime is a tree. Every number is a forest whose trees correspond to the primes its prime factorization. The tree for 2 is a single vertex. The tree for any other prime is a root vertex connected to the forest representing your prime’s place in the list of all primes. (e.g., 7 is the 4th prime, so you create a root vertex and connect it to the representation for 4.)

By user root45:

Each prime number has it’s own tree and we simply put these trees together to represent multiplication. So, once we know what the trees look like for 3 and 7 we can put them next to each other to make 21.

To find out what the tree looks like for a prime, we build it recursively using the number of primes less than or equal to our prime. To start off with, 2 is just a dot (*).

2: *

To make the tree for 3 we note that there are 2 primes less than or equal to 3, so we take the tree for 2 and connect it to another * like so:

3:

*
|
*

To make 4 we first prime factor 4 as 2*2 and then put the trees next to each other.

4: * *

To make 7 we note that there are 4 primes less than or equal to 7, so we take the forest for 4 and connect it to a root.

7:

*   *
 \ /
  *

Now we can make the tree for 21 = 3*7.

21:

* *   *
|  \ /
*   *

So basically you can recursively define each number using prime factorization and the number of primes less than or equal to a prime. Note that the function that gives this number is called π(x) (that is, ‘pi of x’).

Each column contains 3 squares, so each of the 7 columns must contain 2 squares of the same color. There are 3 ways to position 2 black squares in a column, and 3 ways to position 2 red squares, for 6 possibilities overall. Since there are 7 columns in the grid, some 2 of them must bear the same pattern, and thus there are 2 different columns that each contain 2 squares with the same color and position. These form the corners of a rectangle.

(Jeff Hooper, “Pólya’s Paragon,” Crux Mathematicorum 33:3 [March 2007], 146-148.)