Is there an intuitive explanation as to why only the private key can decrypt a message encrypted with the public key? I have just learned about using PGP/GPG for email encryption and one thing bugs me:

How is it possible that a message encrypted with somebody's public key can be decrypted only with that person's private key?

This concept of asymmetric encryption is not easy for me to understand. Simply said, I would expect that the process to encrypt with the public key should be unique and therefore be able to be reverted with the same key.

What's the key operation that ensures this is not possible?
The basic explanation is that you need both keys to make a complete encryption/decryption cycle.

Basically the encryption works with modulo arithmetic so that

$$c=m^a \mod n$$

and

$$m=c^b \mod n$$

where $a$ and $b$ are the public and private key of the algorithm. $m$ is the plain text message and $c$ s the ciphertext.

The most important thing about the formulas is

$$m=(m^a)^b \mod n = (m^b)^a \mod n$$

For this to work $a$, $b$ and $n$ must follow certain restrictions.

This is the basic idea behind RSA and other asymmetric encyption systems whereas you can make the operations more complex/different but the basic idea that using two different keys to complete the encryption/decryption cycle.