Dr. Kreeft puts very succinctly what he calls the Argument from Aesthetics:

There is the music of Johann Sebastian Bach.
Therefore there must be a God.
You either see this one or you don't.

He includes this as #17 of the 20 Arguments for God's Existence. It seems to be a popular argument, and I can see why. It operates on a different level than many other arguments for God's existence. Beauty attracts us. Beauty, inside and out, attracts us to people, to places, and to things.

Why do we make things not just work well but look good? Because beauty draws us in. It attracts us to the goodness that is already there. An appliance may work well, but we like it even better if it's attractive -- if it looks good in our kitchen.  A car that works well is a great thing (and sometimes seems like a hard thing to find!), but a car that also looks good is even better. We feel better when our house is clean or our desk is organized. Why? Because beauty draws us in. It leads us toward the goodness of the thing or the place or the person.

Even if we should simply recognize truth, we have an easier time listening to someone when they look nice. So we dress for interviews and speeches. We fix our hair and check our teeth before a date. We make sure our clothes aren't stained and that we smell nice.  Why? Why not just say "I'm a great guy, who cares if I smell?'  Because we do care!

And so this argument takes that same tack. It starts with beauty, and let's that beauty lead us in toward goodness and truth.

While he isn't talking about this argument per se, this idea is explained well by Fr. Barron on Evangelizing Through Beauty. You can be "overwhelmed by the beauty of a place", he says. If you fall in love with the beauty of a story, or a church, or a piece of religious art, you will keep coming back to that beauty. As you come back again and again, you are automatically "exposed to the moral environment and the religious environment" at the same time.

We move from the beautiful to the good to the true in everyday life.  That's how marketing works. Look at this beautiful product. Now, see how well it works. And now, buy one of your own so you can experience it.  Make it part of your life.

Does this prove that God exists? Not in the same way that other arguments try to prove His existence, no. But this argument, in one form or another, has brought people to faith. As a recent example, Jennifer Fulwiler was moved by the beauty of her child to realize that there must be something more. There are stories of people converted by the beauty of a great cathedral. It's one of the most personal of the arguments for God's existence. I wouldn't say that you either get it or you don't, period; but at any given point in your life, you'll either get it or not. You'll either let it in or not.