English Translation
He watches the news every morning.
I do not understand why the reflexive pronoun “sich” is necessary, unless its purpose is to confuse non-natives speakers!
I think these dative-reflexive constructs can usually be boiled down to a “because I want to”, or “to do myself a favor”, or “to satisfy my interest” as the core meaning.
For example:
Ich kaufe ein Buch. - I’m buying a book. (neutral)
Ich kaufe mir ein Buch. - I’m buying a book (for myself).
The book is bought out of interest. Kind of as a gift to one’s self.
The dative object doesn’t always have to be the speaker themselves:
Ich rücke den Spiegel zurecht. - I’m adjusting the mirror.
Ich rücke mir den Spiegel zurecht. - I’m adjusting the mirror (so I can see properly with it).
Ich rücke dir den Spiegel zurecht. - I’m adjusting the mirror for you (so you can see properly)
Ich mache es mir bequem. - I make myself comfortable. (lit. “I make it comfortable to me”)
Ich mache es dir bequem. - I make it comfortable to you.
Ich mache es bequem. - I make it comfortable. (kind of nonsense, but it would mean “comfortable to anyone who’s possibly going to sit there”)
With “anschauen”, the difference is how actively and consciously you are watching:
Ich sehe das Bild. - I see the picture. (the ability of your eyes to see it, different verb!)
Ich sehe das Bild an. - I’m looking at the picture. (neutral, the gaze is directed at the picture, nothing more)
Ich sehe mir das Bild an. - I’m taking a look at the picture. (almost inspecting the picture, scanning for all the details, actively processing the information in the brain, watching with attention)
In english, there’s some similarities regarding the dative/indirect object that you can find in sentences like “Cry me a river!”, or “Do yourself a favor and get yourself a ticket!”
There’s millions of google hits if you search for “I’ll get me a”, for example: “If I want blind adoration, i’ll get me a dog.”
It’s just that in german it’s not just things you can get for yourself, it’s also actions like watching.
“I’ll watch myself a movie” doesn’t sound right in english, but it’s fine in german. 
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This can also sometimes work in English without the formal reflexive (it’s extremely informal):
I’m going to buy a book. (Neutral: the book could be for anyone)
I’m going to buy myself a book. (Formally correct: the book is for me)
I’m gonna buy me a book! (Very informal, but means the same as above)
“Hey, I heard you just won the lottery. What are you going to do?”
“I’m gonna buy me a car! I’m gonna take me a vacation! I’m gonna have me a good time!”
Even in this clozemaster example, it could conceivably work (though again, it’s very informal):
“He’s gonna watch him some news!”
I know those technically aren’t examples of the reflexive in English, at least not in formally correct English, but it’s remarkable to me how close slang-English can get to German sometimes. When I was a kid, we used to refer to stuff that had broken down as “kaputt”, and I had no idea why until I started learning German many years later.
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I think the additional, optional “sich” reinforces the meaning.
In other words: You don’t just take a (perhaps disinterested) look at something, you take a real close and interested look at it.
Example:
“Er schaute den Baum an.”
Perhaps you take a glance at the tree, from 30 meters afar, through the window glass, because school is just that boring.
“Er schaute sich den Baum an.”
You get real close to the tree, examine it, look and feel the bark and how the wood is formed, count the little bugs and insects crawling over the bark and living in the various holes, etc.
In the example at hand:
Are the news just in the background while you’re making breakfast in the kitchen (without “sich”), or are you actively watching them (with “sich”)?
I know the differentiation is not that clear. You can also express the second meaning without using “sich”. These are just my immediate thoughts.
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English Translation
He watches the news every morning.
Doesn’t Pink Floyd’s “Money” contain an example of this?
“Think I’ll buy me a football team”