Why can't the universe itself "just exist"? Why do we need God as an explanation?

Well let's examine that. If you say that the universe "just exists", you actually are assuming that it existed forever and had no beginning.

But here's the kicker: as far as we know, the universe did begin to exist. Our first proof of that was the development and verification of the Big Bang theory. And recently, in 2003, the Bord-Guth-Vilenkin theorem proved that for more general hypothetical cases, including oscillating universes, emergent models, and any other multiverses that on average expand, these all must have a beginning. Alexander Vilenkin himself said in 2012 that there are no satisfactory models that allow for an eternal universe.

So we're forced to look outside the universe for a cause. Perhaps this video from reasonablefaith.org would help summarize some of the points: Leibniz’ Contingency Argument