My students, when they write essays, they tend to make a statement and then the next sentence is "the reason is".
I give them this example. "Suppose I wrote in an essay 'I hate my mother.' What would you hope is the very next sentence? Right, the reason. A reader will understand your next sentence is a reason for your view."
(The idea of a person hating his/her mother in Korea is so beyond belief that you better have a damn good reason.)
"Now what if I wrote 'I hate my mother. Milk is delicious.' That second sentence does not make sense as your brain expects the next sentence to be the reason. So, writing 'the reason is' is unnecessary."
The reason why we say the reason why, however, I think is generally two fold:
1) it gives us time to think of our reason, or word it. It's an "ummm"
2) making too blunt statements sound rude when spoken. Since English doesn't have formal verb conjugations like other languages, we tend to unconsciously employ longer, more flowery speech when we want to come across as being polite. The unstated idea is if you're spending more time to say something, you're being more thoughtful. It's like the guy at a store who asks the employee with the name tag and uniform "do you work here?" Yes, he's sure, yes it's a stupid question, but it's a way of being unassuming which we then tend to judge as polite.
In writing, it's no doubt sloppy. In speech, it has its effect.