The simplest answer, one which may or may not satisfy, as to why the universe exists is - why not? 
Science doesn't readily answer why questions because science looks at processes, the what questions, the how questions. For instance, the theory of evolution is an idea that explains the processes that produce biological change over time and makes predictions with regard to that change. It doesn't explain how life began in the first place or why it even exists.
The big bang theory explains what happens when we wind the physical processes governing the universe backwards in time. At time=0 we get the big bang, which wasn't really a bang (that is a term coined by Boyle, who fought futilely against the theory until the day he died).

Science doesn't readily answer why questions because science looks at processes, the what questions, the how questions. For instance, the theory of evolution is an idea that explains the processes that produce biological change over time and makes predictions with regard to that change. It doesn't explain how life began in the first place or why it even exists.
The big bang theory explains what happens when we wind the physical processes governing the universe backwards in time. At time=0 we get the big bang, which wasn't really a bang (that is a term coined by Boyle, who fought futilely against the theory until the day he died).
'The difference between a Miracle and a Fact is exactly the difference between a mermaid and seal. It could not be expressed better.'
-- Samuel "Mark Twain" Clemens
"I think that in the discussion of natural problems we ought to begin not with the scriptures, but with experiments, demonstrations, and observations".
- Galileo Galilei (1564-1642)
"In short, Meyer has shown that his first disastrous book was not a fluke: he is capable of going into any field in which he has no training or research experience and botching it just as badly as he did molecular biology. As I've written before, if you are a complete amateur and don't understand a subject, don't demonstrate the Dunning-Kruger effect by writing a book about it and proving your ignorance to everyone else! "
- Dr. Donald Prothero
-- Samuel "Mark Twain" Clemens
"I think that in the discussion of natural problems we ought to begin not with the scriptures, but with experiments, demonstrations, and observations".
- Galileo Galilei (1564-1642)
"In short, Meyer has shown that his first disastrous book was not a fluke: he is capable of going into any field in which he has no training or research experience and botching it just as badly as he did molecular biology. As I've written before, if you are a complete amateur and don't understand a subject, don't demonstrate the Dunning-Kruger effect by writing a book about it and proving your ignorance to everyone else! "
- Dr. Donald Prothero