I'll just pop in to give a brief answer.
Mr. Spock above makes an effort to employ the Big Bang as evidence, so let's start by assuming he grants us all of the science of that bang.
What we know, then, is that there was this single thing which exploded, and that explosion marks the "beginning" of time and space and matter. Since that explosion, we know that natural laws of physics have been the mechanism that caused things to work the way they do and arrange themselves the way they have. In other words, the same science that let us know there was a Big Bang also let us know that, since the Big Bang, everything has come to pass without needing an intelligent hand to guide it (disclaimer: there logically could have been such a hand, but the science works anyway without assuming it).
Thanks to everyday experience (and science too) we know that houses cannot arrange themselves without intelligent guidance. Thanks to science (the science Mr. Spock grants us when he uses the existence of the Big Bang as evidence), we know that the universe can arrange itself, post-Big Bang, without intelligent guidance. Houses and the universe do not analogize perfectly, who knew? (Perhaps we'd like to switch to answering the question, "Where did the wood from which the house was constructed come from?")
As far as the Big Bang signifying the "beginning" of the universe, let me clarify. The Big Bang is an event horizon - we can't see past it (we can hardly conceive of what "past it" would even logically mean). It is the "beginning" of the universe in the same way that, from your vantage point at the top of a mountain, the horizon is the "beginning" of the Earth: it is the most distant thing we can see. Indeed, some have used the analogy that "before the Big Bang" is like "north of the North Pole." It is logically possible that nothing existed before the Big Bang, and it is logically possible that something existed before the Big Bang. We can't see into the "before," so we can't be sure one way or the other. And, indeed, there are plausible models of the universe that operate without a "beginning" in the typical sense.