This use of the genetive case is moreso its absence. It's only really "present" in early writings from around the 3rd millenium BC, and doesn't change the meaning. Essentially, when a sentence speaks of two objects (i.e. where one would expect an "X and Y") where both use the genetive case, the first instance of the genetive can be dropped. Despite being dropped, the meaning remains. For example:
"barag enil barag utu-ak"
literally reads
"Pedestal Enil pedestal of Utu"
But this can be better read as
"Pedestal of Enil and pedestal of Utu"
Let's say we want to translate the phrase "the temple's great fear settles upon the land". It's not the temple that is settling upon the land, but the "great fear" of the temple. I'm not going to go in depth about the verb here, but in Sumerian, the verb must have elements added to it in order to "reference" the nouns. This was hinted to with the ergative/absolutive. I've been told that Basque uses a similar system. So, we're going to want to "reference" the "great fear" doing the settling on "the land" in our verb "to settle". This would mean that "temple" isn't doing anything in this sentence. The "great fear" is just "of" it.
To show this, in Sumerian we do the following:
So, temple is inanimate and 3rd person in this case (the temple's great fear -> its great fear). Let's ignore the nitty-gritty of the verb for now, I'll just put it down. This gives us:
e.ak ni gal.be kalam.a mu.ri.ΓΈ
temple.of fear big.its land.upon settles
the temple's great fear settles upon the land
Remember, when a noun is associated with an adjective, the adjective is the one that gets elements (like case, or possesive pronoun) added to it. That's why the pronoun "be" (its) is on "gal" and not "ni". Another (more literal) way of translating this would be "Of the temple, its great fear settles upon the land". As you can see, "of the temple" is on the far left, hence the name, "left dislocation".
Left dislocation genetives can also be stacked like how regular genetives are. Let's show this with another example I found from a grammar:
"She will tell to you the holy star of the building of the temple" (She will tell to you the temple's building's holy star)
e.ak du.be.ak mul kug.be.a [gu ma.r.a.de]
temple.of building.its.of star holy.its.to [tell]
You could also translate this as: "Of the temple, of its building, to its holy star she will tell to you."
Why is left dislocation used?
It allegedly shows importance/emphasis. In our above example, we're being told of the temple's thing, something "of" the temple. By using left dislocation, we're putting the temple first. That way, the reader quickly knows that what we're talking about is going to be related to something associated with the temple. It ain't just any holy star, it's the temple's building's holy star.