Evidence for Creation

There are a number of lines of evidence that are so strong that there is no serious doubt left in astronomer's minds that the Universe essentially sprang into existence (in its current form, anyway) about 15 billion years ago. At that time its scale was vastly smaller, and its density and temperature vastly higher. In addition to the Hubble expansion, there are 2 other facts which have no other easy explanation.
One is the presence of the cosmic microwave background (CMB). This is relic radiation left from the Big Bang itself - essentially an image of the fireball when it was last opaque. Because that fireball was the Universe itself, it is visible in every direction. The essential point is that this radiation is an almost perfect blackbody spectrum, and you know that to produce that you must have a hot, opaque source. It seems inescapable, therefore, that the Universe was once hot and opaque. It was predicted before the CMB was observed that such radiation should have cooled off by now to a few degrees above absolute zero. Another way to look at this is that the CMB comes to us from near the observable horizon, which is near the beginning of time, and the Hubble expansion will redshift it down from the ultraviolet (since the Universe was about 10000K when it became transparent) to the microwave part of the spectrum (corresponding to 3K). The Universe changes from opaque to transparent when electrons can remain bound to protons and react only to photons which match the energy levels of hydrogen; before that the free electrons could scatter all photons and there was an electron "fog". As the photons found themselves unimpeded they flew freely until we detect them today, so they give us an image of the Universe at the "recombination era", about 300,000 years old.
The other evidence for creation is that this Big Bang scenario allows us to calculate the state of the Universe all the way back to a tiny fraction of a second after it starts. Its temperature and density begin extremely high, then drop in a known way. A few minutes after creation, the average location in the Universe looks like the interior of a star. Hydrogen fusion takes place, and begins to create helium (and a little deuterium, lithium, and beryllium). The observed abundances of these elements in the overall Universe matches the predictions of this model in detail, and so constitutes strong evidence in favor of it. We can even deduce what the density of normal (baryonic) matter in the Universe must have been (and so must be now) by the ratio of deuterium to hydrogen and helium; this matches well with the total measure of normal matter by very different methods.

The Big Bang

The basic process in the Big Bang is the free exchange of mass and energy, according to E=mc². When the temperature is high enough, the E of the photons exceeds the mc² of various particles. Two such photons can meet and spontaneously change into the particle and its antiparticle. Each of those can then meet its opposite, and annihalate back into two photons. But the Universe is expanding and the photons are steadily losing energy due to the expansion of space. When the energy of the photons drops below the critical limit, they no longer produce particles. All the particles then around will meet antiparticles and change back to photons, which end up cooling. You would think that this should finally leave no particles. There was apparently a tiny asymmetry in the production of particles, so that an extra amount of matter over antimatter was made. We can figure out how much this was by counting microwave photons today compared to protons (for example). It turns out that only 1 out of every 100 million pair productions had to produce no antimatter to account for all the matter we see today (assuming all antiparticles took themselves out with particles and left photons). Such an asymmetry can be produced in certain theories, after all the Universe is itself asymmetric (expanding rather than contracting - time only moves forward not backward).

In this way all the known particles "freeze" out of the radiation field as it passes critical temperatures; the more massive particles freeze out sooner since they require hotter photons to make them). It only takes about 10 seconds to produce all the matter present today. Then in the next few minutes, the only heavier nuclei made in the Big Bang appear (largely helium). After that, for a few hundred thousand years, the Universe is a hot expanding fireball with no atoms. The atoms recombine when the temperature drops below about 10000K (the temperature of an HII region) and the radiation becomes fully decoupled from the matter (and the Universe goes from opaque to transparent). It continues to cool until the peak wavelength moves longward of the optical range; the Universe then goes dark to human eyes (though of course there are none then). It doesn't light up again until gravity has worked on the matter long enough to begin collapsing it into stars, clusters, galaxies, and clusters of galaxies (after a billion or two years).

Inflation

We now think that even more bizarre events went on before the first particles froze out. At about 10^-35 seconds after the beginning, we believe that the scale of the Universe went through a huge inflation (perhaps by a factor of 10²⁵). The reason behind this is the phase change of the Universe from a state in which there is symmetry and all the forces are the same, to the current state. The previous "false vacuum" becomes a true vacuum with negative potential energy due to its expansion, and positive mass-energy initially in the form of the hot radiation field. The sum of these two can remain zero. The equilibrium before inflation establishes the conditions that will prevail after. This solves the "horizon problem", which is that the CMB looks the same in all directions, but the matter in one direction cannot know what temperature the matter in the other direction has, because they are well outside of each other's horizons. The Big Bang theory by itself has to resort to saying they just happened to all be made at the same temperature, but inflation provides a physical reason why they are the same. It also predicts that we should see structure in the CMB which is so large that it too cannot be fully causally connected (one end is outside the horizon of the other end); such structures are stretched to that size by inflation. It was a great triumph of the theory that the COBE satellite actually saw such things - they are very hard to explain without inflation. One effect of inflation is that whatever the curvature of the observable Universe was before inflation, it will have to be flat afterwards. This is analogous to the balloon in the lecture room - no matter what shape it has there, if I suddenly blow it up to the size of the Earth the part of it that is still in the room will look flat. This solved the "flatness problem", namely that the Universe has increased its scale by at least  a factor of 10⁶⁰ since matter appeared. We can measure the density today, and it is reasonably near the critical density (required in a flat Universe). It therefore had to be within 10^-60 of being flat at the beginning, so that's essentially flat!

A summary of the History and Fate of the Universe