This is indeed a good question; let me try to make it a bit more precise. Suppose:
- Alice has a plaintext message of some number of bits, call it p.
- Alice and Bob share a crypto-strength random number generator that generates n truly random bits.
- Alice and Bob share a pseudo-random number generator that can take a seed of size n and produce one of 2n sequences of p bits.
- Alice and Bob have an insecure channel and a secure channel.
- Alice (or Bob) creates a truly random key KEY of size n and sends it to Bob (or Alice) over the secure channel.
- Alice creates a pseudo-random sequence SEQ of size p and xors it with the plaintext to produce a ciphertext of size p
- Alice sends the ciphertext to Bob over the insecure channel.
- Bob decrypts the ciphertext in the obvious way.
- Details of the system are not part of the key. That is, the attacker should be presumed to know precisely how the PRNG works.
Ok, so first of all, the obvious criticism is: if Alice and Bob have a secure channel then why are they messing with encryption at all? And the answer is: the secure channel may be more expensive or only available at inconvenient times.
The second obvious criticism is: if Alice and Bob have a device that can generate n truly random bits then why not use it to generate p truly random bits in the first place, and get the pseudo-random number generator out of the picture? And the answer is: because sending p bits over the secure channel may be too expensive.
So what attacks could be mounted against this system?
The accepted answer points out that this system only defends against one attack, namely, it makes it very hard for Eve to decrypt the intercepted ciphertext. It does not provide any mechanism for Bob to verify that the ciphertext received was actually produced by Alice. If bits of that ciphertext are flipped, they'll be flipped in the plaintext too and Bob has no way of knowing that.
Another attack is: suppose n is a relatively small number, say, 32. That means there are only four billion possible values for KEY and therefore only that many pseudo-random bit sequences SEQ. Eve can obtain the ciphertext and simply try all of them until one of them decrypts it into something sensible. This implies that n had better be pretty large; large enough that this brute-force attack is infeasible. (And of course, if n has to be so large to prevent this attack that it is a significant fraction of p, then again, take the PRGN out of the picture and just generate as many truly random bits as there are bits in the plaintext.)
And yet another attack is: suppose the attacker manages to obtain the whole ciphertext -- easy -- and manages to obtain or guess k bits of the plaintext. This is not as hard as you might think; lots of messages have common patterns in them. (WWII era allied codebreakers made good progress by assuming that a significant fraction of messages would end in a greeting to the Führer; and they were right.) If k bits of the plaintext and all of the ciphertext are known then k bits of SEQ are known. Now the question is: Can the attacker make a good guess as to the value of KEY given k bits of SEQ and knowledge of the internals of the PRNG? If they can then the task of generating SEQ just got a lot easier for the attacker. The PRNG had better be carefully designed to make this kind of key recovery infeasible.