Since the RSA algorithm for public/private key algorithm can only encrypt payloads as large as the (public)key, this is the only way to safely encrypt files.
It's not the RSA key size that prevents you from using RSA for arbitrary message lengths. Notice that block ciphers used for bulk encryption are similarly constrained by their block length (which are far shorter than an RSA key). In both RSA and AES, you can use a chaining construction to encrypt arbitrary-length messages.
The things that keep you from doing this in the real world with RSA are security and speed.
It's first of all much slower to perform a single RSA operation than it is to perform the AES block transform. AES involves no bignum math at all, let alone bignum modular multiplication.
Secondly and more importantly, RSA encryption is fundamentally volatile and dangerous. As an exercise, go implement it in Python or Ruby (you're going to find it's remarkably easy, since both those languages automatically promote to bignums). RSA is just a simple formula. As a result, there are a variety of pitfalls to using it safely. Among the important ones is the fact that you can't safely encrypt related messages, and that messages require a certain proportion of random padding.
That's why PHP provides the openssl_seal function can do exactly the same (and more) as the given code. (http://php.net/manual/en/function.openssl-seal.php)
When you Google for it there are also implementations for other languages available.