That version of the Cosmological Argument simply points to the universe having a cause. The Christian response would be that the cause is God. Science confirms that the universe must have had a beginning. The Christian view is that something that does not need to be given existence must exist to give everything else existence. That something is God. God does not need to be given existence because God is the cause who exists to give everything else existence.
I found this version of the Cosmological Argument (36 Arguments for the Existence of God by Rebecca Goldstein, published in 2010 by Pantheon Books, a division of Random House Inc.) which defines it this way:
1 – Everything that exists must have a cause
2 – The universe must have a cause (from 1)
3 – Nothing can be the cause of itself
4 – The universe cannot be the cause of itself (from 3)
5 – Something outside the universe must have caused the universe (from 2 and 4)
6 – God is the only thing that is outside of the universe
7 – God caused the universe (from 5 and 6)
8 – God exists
Not only did the universe have to have a first cause to get started, it needs something to give it existence right now. The only thing that would not have to be given existence is a thing that exists as its very nature. It is existence. This something would always exist, have no cause, have no beginning, have no limit, be outside of time, and be infinite. That something is God!
This article explains it far better than I ever could: What is the Cosmological argument for the existence of God?