How did the universe come into existence?

No one knows for sure yet.
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The most honest response is one of ignorance. No one knows. Existence itself is a paradox. All cosmological arguments only go back so far, but one can always ask for another cause. The prime mover, or the causeless cause, is one that has been questioned since Socrates. Atheists criticize theists for the same type of logical arguing that they themselves are guilty of. A deity makes a perfect causeless cause. Or does it? Whereas a deity may explain the existence of a universe it does not explain the existence of a deity. But, deities that create existence do not exist as a part of the existence they create... so one can easily say that the deity both exists in some fashion and does not exist in some fashion. The rules of logic and of physics and the demands of science are not applicable to a creator deity because that deity creates physics and logic along with everything else. Without a deity, on the other hand, we are still left with a question of cause. All science, no matter how far back it traces existence, requires something before it... something to have caused it. The problem with the scientific approach is that it can never escape the confines of the material that science treats, and is therefore forever doomed to fail in answering the question fully. The irony is that the more science attempts to answer, and the more complicated the subject matter, and the further back in time we extrapolate... the less confirmable the theory becomes, the more convoluted it becomes, and the less comprehensible it becomes. Something about Occam's Razor comes to mind. Even when science explains the behaviors of matter and energy it still has to justify the laws themselves... do the laws of physics emerge as a consequence of the behaviors of matter and energy or does matter and energy exist and behave because of the laws of physics? Which comes first and why? Even arguments from pure logic that boil down to "existence necessitates itself" rely on the prerequisite that logic must innately exist, that logic in its purest form has the power to compel existence. Either way you want to go, faith is an integral part of knowledge. We must put faith in science and its methods, as well as the integrity of those performing it, and premising this we must put faith in the logic we reason by, the philosophies that underpinnes all of the above, etc., etc. There are more enumerations and compounding steps of faith-full belief between a few observations and conclusions about existence or about deities. But believing in a deity straight away is more direct, and no one makes the mistake of denying their faith on account of being too confident in their science. We criticize emotional-based rhetoric but the truth is, all faith and in turn all sense of certainty and all confidence in knowledge, all determinations of "rational or not", boils down to a very primal and evolved (or designed) sense of confidence and feeling about whether or not one thought makes more or less sense than another. Why does a contradiction prove an argument false? Because I say so? Because it doesnt "feel right"? Or because it is inconsistent with an arbitrary set of man-made axioms about how logic works? Without the need of me pointing out that contradiciton occurs all the time in nature when you view the quantum world!