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The argument runs as follows:
1) Things in the universe (U) are contingent (C)
2) Anything C depends on a Necessary Being (NB) to make it the way it is
3) Things in U depend on NB to make them the way that they are
U: Anything that exist apart from NB
C: Something that needs to be made the way it is
NB: Something that does not need to be made the way it is
The things in U are C, since they need something to make them the way they are. Science is a manifestation of the search for explanations for C, since it attempts to show what makes things the way they are. The ultimate thing that C depends upon cannot be within the universe, since it would then exist in space-time, which is C.
Analogy: you hear that someone has been spreading a rumour about you, so you ask him, but he says he heard the rumour from person 2. Person 2 says he heard it from person 3 etc. Obviously this rumour has to have a source, someone who made it up without hearing it from someone else. If there was an infinite series of people without someone who initiated the rumour, there would be no rumour for the infinite series of people to spread. The number of people is irrelevant. If there are 5 people, 20 or infinity people, unless there is an initiator for this chain, no rumour exists. Applying this to contingency, unless there is something to make things the way they are and which doesn't need to be made the way it is, nothing would be the way they are.
Side points: This argument is not based on causality.
Round 1
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Round 2
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Round 3
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Round 4
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Round 5
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What is necessary being? Because if it's just "something contingent things depend upon", then this argument is not necessarily about God.