Why use complex encryption algorithms? I have recently gotten interested in cryptography, so I looked some things up.

I thought, why even use AES or DES or any other complex way for encrypting data when there are far simpler ways, like just generating a random (pseudorandom) bitstream, and XOR'ing it with the data or secret message? The point is, if you have a random enough bitstream, you can just XOR the data. 

Of course, you're the only one who will know the stream.

So why use all sorts of encryption if you have these simpler, easier ways of doing things? I'm new to cryptography, so this question may sound a bit naive.
We use more complex encryption algorithms than XOR  with a random or pseudo-random keystream for a number of reasons:


In order to get a short secret key in symmetric encryption. XOR with a true random stream (One Time Pad) requires storing or/and transfering a secret keystream the size of the data to encipher, which is utterly impractical. Replacing the keystram with a pseudo-random stream (as outlined in the question) solves that key size problem, but see below.
In order to construct Cryptographically Secure Pseudo-Random Number Generators then used as suggested in the question. A weakness in a PRNG could become a weakness in the encryption scheme using its keystream, especially if that's by XOR. We know several demonstrably secure ways to turn a secure block cipher into a CSPRNG, that are reasonably efficient, and (thus) popular.
In order to get direct access to encrypted data. Imagine a large movie stored encrypted as proposed in the question. In order to start viewing it in the middle, we have to run the pseudo-random generator from the start of the file. That's solved by encryption using a block cipher (such as AES) in most common modes, including CTR/CBC/OFB/CFB. For example, AES-CTR does as in the question, with the pseudo-random stream obtained by enciphering a counter (starting from an Initial Value usually stored at start of file), so that we can produce the keystream for any position by adding the IV and the (appropriately scaled) file index, and enciphering the counter values startign from there with AES, yielding the keystream, which allows decryption starting at the desired point.
In order to also get assurance of data integrity and origin, which often turns out to be at least as needed as data confidentiality. Even if we assume a large secret keystream, we need something slightly more complex than XOR with the keystream (e.g. universal hashing) in order to ensure that an adversary did not mess-up with the data (e.g. change "pay \$100" to "pay \$900").
In order to get asymmetric encryption, that is the capability to encipher without needing anything secret (beyond the data to encipher) on the side that enciphers.