It is explained in the article you linked ...
you can create environment variables with specially-crafted values before calling the bash shell. These variables can contain code, which gets executed as soon as the shell is invoked.
Which means the bash that is called with -c "echo this is a test" executes the the code in the single quotes when it is invoked.
Bash has functions, though in a somewhat limited implementation, and it is possible to put these bash functions into environment variables. This flaw is triggered when extra code is added to the end of these function definitions (inside the enivronment variable).
Means the code example you posted exploits the fact that the invoked bash does not stop evaluating this string after performing the assignment. A function assignment in this case.
The actually special thing about the code snipped you posted, as I understand it, is that by using putting a function definition before the code we want to execute, some security mechanisms can be circumvented.