Decrypting bit stream encryption where the key is known

I am writing a chapter for a book on Security and am working my way through the history of encryption. I know enough about encryption to know what I can and can't crack and the different techniques of cracking (dictionary attack, amn in the middle, etc) but I have never until now spent a lot of time learning about HOW the encryption actually works, or the nuts and bolts of cracking it. Now, I could sit here and tell you what a math whiz I am but I'd be lying.  What I am trying to do, in my section on Stream Ciphers, is demonstrate how if you use the same exact random stream cipher twice on two different messages, the message can be compromised.  I have read this in multiple articles about stream ciphers, that it is very easy to do, but 4 hours ago I sat down to try to do it and after lots of equations and crumpled up paper, I'm getting dizzy.  

Can anyone explain to me, in plain english (OK, you can use the word XOR if you really need to) how to go about cracking it?  What techniques?  Chi-square? Freq. Analysis??  If you are ok with it, I will use your explanation in the section on cracking random stream ciphers and reference your name at the end of the chapter.  Otherwise I will just restate it in my own words.  I have attached the section, it is still pretty tentative and rough, I am not certain that the ciphertext is susceptible to frequency analysis or not.  I am trying to work out how it would be cracked.  Start Free Trial