Question

According to the Big Bang Theory, how long ago did the universe expand explosively into existence

Answer 1

According to some of the same evidence that is explained by the "Big Bang" theory, the universe expanded into existence roughly 13.7 billion years ago.

The Big Bang theory says nothing about the conditions at exactly the instant of the "beginning" or less than about 10⁻⁴³ second after it, and nothing about any "before".  

In fact, it says that anything "before" is "unknowable", and in some sense, that 'Time' itself started with the 'big bang'.  

Answer 2

About 13.7 billion years ago

The Big Bang Theory states that the universe started about 13.7 billion years ago, and before that, everything was in 1 singularity.