How can the universe exist?
We can say two things about this question.
The first is that we are observing the universe, so we have to conclude that it exists. This is the base of the scientific process, the experiments and observations are the base of our knowledge.
So a possible answer to "How can the universe exist?" is because we can observe it.
The second answer is: ok, we see it but how do we explain its existence?
The answer to this question is not easy because we basically we do not know all the process that originated the universe.
Today we know that the universe is expanding, so we imagine that in the past the universe was smaller than today.
Actually all the mass of the universe was collapsed in a volume small enough to destroy any atomic structure. Not even the particles were there because the energy's density was too high. So all the matter that we see today was condensed as pure energy, in a form of radiation.
So the universe was composed only by radiation and then cooled down generating the masses of stars, galaxies etc. We know that this theory is valid because not all of this radiation transformed in mass and we can still see it today. It is what we call cosmic background radiation (or microwave background radiation).
Ok, but what generated this radiation?
We know that the "radiation" period was 20.000 years long with an initial phase very dramatic in acceleration (we call it inflation period).
But we do not know very well how it was the universe at the beginning of this period.
Scientists refer to the initial moment as Big Bang.
The problem in understanding what the Big Bang was is that even the time was generated by it. This is not easy to understand because it means that the question "ok but how was the universe before the Big Bang?" does't have a meaning.
To conclude: what we know today is that somewhere in the past there was a strange event that we call Big Bang, this produced, space, time and all the energy that we see today. This energy was in a extremely dense state for 20.000 years as radiation, until the universe was big enough and the energy density decreased enough to cool down. The radiation then started to transform in particles, electron protons, etc. The matter combined together in atoms producing mainly hydrogen. Hydrogen condensed in big masses because of the gravity and started the nuclear fusion.
These are the stars. The fusion produced heavier atoms (helium, lithium... till iron) and neutrons. The neutrons captured by atoms produced new atoms heavier than iron up to the unstable atoms (like uranium). And the aggregation of these atoms produced planets.