Suppose you saw a ball in mid air. Balls don't just hover in mid air, so if you see a ball in mid air you know that someone must have thrown it. You could do things like measure the ball's velocity, and this would tell you whether the thrown ball is on the way up or on the way down. Or if the ball's velocity was upwards and very high you could tell if the ball was going to escape Earth's gravity never to return.
This is the situation we find ourselves in when we observe the universe. We see a large number of stars, galaxies etc, but they can't just be hanging there because that isn't a stable configuration - they must be moving. And when we measure the velocities of galaxies we find they are all moving away from us, so the universe is like the ball on its way up.
Knowing the current positions and velocities of all the galaxies we can extrapolate backwards in time, and when we do so we find that as we work backwards in time the velocities get higher and higher until 13.7 billion years ago they become infinite. That is of course the moment of the Big Bang.
The analogy with the thrown ball is actually pretty close. At the instant of the Big Bang the galaxies all started out with very high velocities, just as the thrown ball started out with a high velocity, and for the first few billion years the mutual gravity of all those galaxies was slowing them down, just as the ball slows down as it moves upwards. If dark energy didn't exist the galaxy velocities would continue to decrease. We think the universe is flat, which is analogous to the ball having exactly escape velocity, so the galaxy velocities would tend asymptotically to zero given infinite time.
The problem is that we know why the ball started out with a high velocity because someone threw it. But we don't know why the galaxies all started out with high velocities because our theories don't go back in time far enough to describe the Big Bang. We hope that some future theory of quantum gravity will explain why the universe started out as it did 13.7 billion years ago.