The top scientific theory regarding the beginning of the universe is the Big Bang, a term first used by Fred Hoyle in 1950. In this theory, the universe began 13.7 billion years ago. It started as a single point, smaller than an atomic nucleus, with an infinitely high temperature and an infinite density. In the first instants of time, between 10-35 seconds and 10-32 seconds, it underwent a period of exponential "inflation", doubling in size every 10-24 seconds, before settling down into the rate of expansion we observe today.
A common misconception is that the Big Bang was an explosion that happened out in space. The Big Bang did not happen in a particular place, as space only came into existence with the Big Bang. Space itself was created and then stretched; taking all the matter and energy that makes up the universe with it. In the same way, there was nothing before the Big Bang as time also started with the Big Bang: there was no "before" for anything to be happening in. Nobody has come up with a testable explanation of what caused the Big Bang, and the question may not actually be meaningful. This is mind-boggling stuff, but it does seem to fit the facts.
There is good evidence for the Big Bang. The theoretical foundations were laid by Einstein's theory of General Relativity and a concept called the Cosmological Principle. The observational evidence comes from three things:
When you look at stars and galaxies in the night sky you are looking back in time. The further away an object is, the longer it has taken for its light to reach us. The furthest back we have been able to see is to about 400 million years after the Big Bang. This was about the time the universe had cooled sufficiently for protons and electrons to combine to form hydrogen atoms. Even with the biggest super-telescopes being planned, we will never be able to see right back to the beginning of time for the simple reason that there was no light to see until around 380,000 years after the Big Bang. This is like an optical curtain at the edge of what cosmologists call the Dark Ages, before which there were no stars or galaxies emitting electromagnetic radiation for us to detect.
Despite this, there is plenty of reasonably direct evidence for how the universe unfolded from about 1/100th of a second after the Big Bang. Before that, things were very different and strange. In those earliest moments, all the elemental particles and the forces that determine how they interact must have come into being. Understanding what happened will give us a much better idea of how the universe came to be the way it is today, with this particular set of particles, forces and physical laws.
Over the last century or so physicists have developed ideas about the basic physical features of this early period, but important details are still the subject of intense scientific speculation. Putting these ideas to the test is a major part of the work of the LHC, achieved by creating particle collisions that are so energetic that they recreate the sort of conditions thought to have existed just 1/100th of a billionth of a second after the Big Bang.
It's the next best thing to actually witnessing the beginning of the universe!