"The girl's cat"
(Info under 'Possessives' -> Tips on Website) 3rd person, singular & plural, the suffix added is -(s)I So Kizin = of the girl's, then: Kedi -> cat + si = Kedisi -> "cat that is possessed"
So, objects that are possessed by a third person receive the '-(s)I' treatment. Thus making the sentence. I hope this was clear!
The -lar ending is for more than one of something. So kızları would be saying many girl's cat. -(n)In ending is specifically showing possession. The girl, kız, owns (possesses) the cat, so we have to add the suffix making kızın.
Once you have -(n)In in a suffix the direct object (the thing possessed) needs a special ending too. Because if something is possessed, then that something HAS to be acted upon.
And to show that we add the -(s)I ending, not the -(n)I. The -(n)I ending is for having something. The cat doesn't have anything. But it is a SPECIFIC cat, and it is 3rd person, so it must take the -(s)I ending.
You can read a better explanation in the Tips section of this unit.