Is there an intuitive explanation as to why only the private key can decrypt a message encrypted with the public key?
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.