How do you explain to yourself the existence of this Universe or the existence of anything at all?

Asking why is meaningless, it's an artifact of how our brains are wired.

Our brains are evolved to try to understand the world around us in terms of reason; if this happens then that happens. It makes sense when looking at the chain of results that cascade out of the fireworks of creation; chemistry, biology and physics. The arrow of time points one way and we have evolved around that premise.

It doesn't mean that our intuitions are correct or even vaguely headed down the right path. We're a victim of that arrow of time, the path of the world we live in.