The meaning depends on context. Taken literally, it means to destroy completely using explosives. Metaphorically, several meanings are possible. It can mean to refute an argument completely or to reveal a secret.

To add to Randy's answer - yes, it normally refers to literally blowing something up with a bomb/dynamite, etc. Often, I see it used to mean "to reveal something so amazing/overwhelming/unexpected that you will feel like you have been blown up by dynamite". This could be something like a secret, but more likely a new car, shopping centre, idea at work, etc. It is also very often said as "I will blow your minds". Tom