The operation at the core of RSA is a modular exponentiation: given input m, compute me modulo n. Although in general this is a one-way permutation of integers modulo n, it does not fulfill all the characteristics needed for generic asymmetric encryption:
If e is small and m is small, then me could be smaller than n, at which point the modular exponentiation is no longer modular, and can be reverted efficiently.
The operation is deterministic, which allows for exhaustive search on the message: the attacker encrypts possible messages until a match is found with the actual encrypted message.
The modular exponentiation is malleable: given the "encryption" of m1 and m2, a simple multiplication yields the encryption of m1m2. This is akin to homomorphic encryption, which can be a good property, or not, depending on the context.
For these reason, the integer m which is subject to RSA must not be the data to encrypt alone, but should be the result of a transform which ensures that m is "not small", contains some random bytes, and deters malleability.
The "v1.5" padding in PKCS#1 does the job reasonably well, subject to two (known) caveats:
A decryption engine can be turned into a padding oracle if the attacker can submit malicious messages to decrypt, and observe whether the decryption engine found a correct padding structure or not. This is Bleichenbacher's attack, which could work against SSL server, requiring about one million aborted handshakes to recover the private key. SSL servers have since been patched (if decryption fails to find a padding, the engine nevertheless continues with a random blob instead of the "pre-master secret"), but this highlights a potential problem with PKCS#1 v1.5.
This padding scheme was never "proven". Security proofs are a neat thing to have. In my opinion, this padding got something better, which is that it has been deployed in the field and widely used for almost 20 years; but many people prefer it when there are some mathematics which corroborate experience.
OAEP aims at making things better on these two points. It turned out that for padding oracles, things are not that clear; and the first proof was shown to be somewhat wrong by Shoup. The proof was "repaired" by Fujisaki, Okamoto, Pointcheval and Stern in the case of RSA (OAEP was designed as a generic padding scheme, but we now have a proof of security only when used with RSA).
To sum up, OAEP tries to improve over the previous padding for resistance against chosen ciphertext attacks (not chosen plaintext attack: since the public key is public, anybody can encrypt anything at will), but the v1.5 padding was already heuristically good at it, up to a million or so decryption, which is good enough for many purposes, and can be patched for others (as was done for SSL). OAEP does better if correctly implemented, which is not as easy as usually believed.