Allow me to start with the basic outline of the argument, I will then go on to explain how this argument necessitates the conclusion, offering definitions along the way.
The Cosmological Argument:
P1: Everything that begins to exist has a cause.
P2: The universe began to exist.
C: The universe therefore as a cause.
This is generally considered a relatively fundamental law of causation . Changes in state (going from not existing to existing) require causation. We should consider that any effect that lacks a cause becomes, by definition "necessary." And self sufficient effects cannot, by definition "begin."
In the past, some have sought to object to this premise by forwarding different aspects of Quantum Mechanics. These fail however because the causal mechanism still exists, it is the quantum wave function . The confusion often arises because we confuse a probabilistic cause for no cause at all. If there was a random number generator that killed a cat on odd numbers, we wouldn't say that the cat's death was uncaused.
This premise also is generally scientifically accepted. Inflationary cosmology dictates that the universe began from a near singularity. I think it is important here to point out that time is a physical dimension of our universe, just like the other dimensions. Just as they expanded from a singularity, so did the temporal dimension of our universe. This necessitates a beginning of the universe when the temporal dimension was a singularity as well.
Objections to this premise are usually in the form of alternative hypotheses about our current universe. Historically, the steady state universe was used. That is to say, it was argued until recently that the universe is eternal, that it had always been. This is problematic for several reasons. Primary amongst them is the evidence indicating the universe is expanding. It is for this reason that virtually no cosmologist holds to steady state theory today. The historic objection also still holds. If the universe was eternal, we would expect that all the stars and galaxies to have burned out by now. If there is an infinite past, an infinite amount of time would already have occurred, which is far greater than the possible time limit on all the fission of all the matter in the universe.
The first modification of this theory to deal with the expansion of the universe came with the cyclic model. In which the universe expands, collapses and expands again. This theory however fails because it also cannot recede into the infinite past. Entropy between cycles would build up causing later cycles to be high entropy states and prohibit matter and star formation. Again, if the universe were infinitely old, this would have already occurred and we could not observe star formation now.
Finally, the most modern objection arises from an appeal to a multiverse or multiple universes. This objection also fails for two reasons. One, since it produces a temporal effect, the multiverse itself would need a temporal component (non intentful causes cannot act outside of a dimension they exist in), making it open to the same appeals to an infinite past that we have above. Two, a multi-verse hypothesis would need to be reconciled to the Borde-Vilinken-Guth Theorem  which prohibits low entropy, expanding universes (ie the kind we live in) from any multiverse. To date, no reconciliation has been put forward, with Stephen Hawking noting that this is the single greatest objection to his views.
It naturally follows from the premises that the universe therefore had a cause.
But we can go a little bit further than that. Given the established premises and conclusions and some other observed facts, we can reason out a few of the properties of this cause.
1) Omnipotence. This word is often used in a differing manner than how theists intend it. It does not mean, for example, the ability to do anything such as creating a round square. Rather, when used here it refers to the ability to actualize states of affairs. I will borrow William Lane Craig’s definition here:
Rather we should think of omnipotence in terms of the ability to actualize states of affairs. A state of affairs is just a way something might be – for example, the state of affairs of there being chairs in this room, or the state of affairs of our being in the lower story of the church building, or there being a piano here. Those are all states of affairs that actually obtain. Omnipotence should be understood in terms of the ability to actualize states of affairs. To be omnipotent means the ability to bring about any state of affairs which is logically possible for any one in that situation to bring about.
This ability is a natural conclusion to the CA as I have presented it. In order for a cause to be sufficient to cause the universe, it must be able to actualize states of affairs related to all the specifics of our universe. It must be able to affect physical laws, physical constants, and dimensionless constants. This ability fits the definition proposed above as omnipotent.
2) Aphysical and atemporal. Both of these terms mean that the item in question lacks physical and temporal characteristics. Given that both time and space are properties of this universe and that an effect cannot be its own cause (a logical paradox), we see that the cause defined in our conclusion cannot exhibit properties of its own effect. Given that it must be transcendent of this universe (ie it cannot be bound to this universe otherwise it couldn’t exist to elicit the effect) it cannot be limited by the dimensions of this universe.
3) Intentfulness. This conclusion arises from the observed temporal finiteness of the universe. We know that the cause cannot be a mechanistic cause (IE if the cause exists the effect exists) because we can describe a state of affairs where the cause exists, but the effect does not. This is really a long winded method of saying “the universe began.”
Likewise, we can say that the cause is not a probabilistic cause either. Probabilistic causes require a dimension to act along. IE along a temporal dimension (chance over time) or a physical one (chance over distance). However, all probabilistic causes must act along the dimensions that they elicit effects within. IE, a quantum wave function acts along a temporal and physical dimension to create an effect in both (a particle’s location). You cannot have a quantum wave function (or any other probability function) that only discusses time, but produces a physical effect.
Given now that we’ve ruled out those two methods of causation we are only left with intent. Only a cause that has an intent can demonstrate the attributes labeled above. Only an intentful cause can create information that is not found within itself. IE all causes except intentful ones have temporal information within them if they act temporally, physical information within them if they act physically, etc. Only intentful causes exhibit the kind of causation we observe given the CA.
So we can see that given the premises that the universe must itself have a cause and that this cause must be aphysical and atemporal since it cannot be part of its own creation, that it must be omnipotent in order to create that creation and that it must be intentful in order to explain the finiteness of the universe and its dimensionality.
Given the premises, which are supported, no other conclusion can be accepted.
Now for a miscellaneous definition:
Logical necessity: I don’t mean this term to imply philosophic necessity in that I argue that no other belief is possible, but rather rational necessity in which I hold that no other conclusion is rational.
5) I. D. Novikov and Ya. B. Zel’dovich (1973) Physical Processes Near Cosmological Singularities Annu. Rev. Astro. Astrophys. 11 387-412
S. W. Hawking and R. Penrose (1970) The Singularities of Gravitational Collapse and Cosmology. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 314. 1519. 529-548.
6) Arvind Borde, Alan H. Guth, and Alexander Vilenkin (2003) Inflationary Spacetimes are Incomplete in Past Directions Physical Review Letters 90. 15 http://arxiv.org/pdf/grqc/0110012.pdf