It may not require knowing how encryption works to know how things would be affected, if all you knew is that a technique for easily finding prime numbers makes modern encryption techniques easy to crack. McCanney says that using his method, a third grader can crack what the NSA cannot. I find cryptography to be more boring and tedious than playing cards (I can't stand playing cards) but here's a breakdown anyway of how it works:
Two random prime numbers are chosen to make up the secret key. They are multiplied to get the public key. The encryption algorithm uses the public key to garble the message in such a way that only by knowing the secret key (two primes that were multiplied) can the message be decrypted. If all you know is the public key, you would need a computer that can go through a bunch of prime numbers and test if any of these evenly divide into the public key to get the secret one.
So let's say 53 and 191 are the secret key. They are multiplied to make 10123, the public key. The cracking computer would divide 10123 by various primes, and soon it would find that the prime number 53 divides into it cleanly, so then 53 and 191 could be used to decrypt the message.
However, in modern encryption the secret and public keys are made of numbers so HUGE that the computer wouldn't find a prime that worked until it got up into the septillions or even larger instead of mere 53 or 191, and that would take practically forever. And so the public key stays safe.
That prime numbers are thought to be distributed randomly on the number line is what requires so many numbers to be tested, since there is no hint about where to start. But knowing the order behind primes would allow the entire process to be shaved down to nothing, and so a public key could be easily and quickly decomposed into its secret key and the message decrypted.
Acquiring fringe knowledge is like digging for diamonds in a mine field.
J