The aesthetic argument for God's existence is an intriguing subset of the teleological (design) argument. Think of the following:
1. Music displays simplicity in diversity. (Premise)
2. Simplicity in diversity is beautiful. (Premise)
3. Therefore, music is beautiful. (From 1 and 2)
What makes a song so aesthetically pleasing? If it were all the same (simplicity), it would be boring. If it were all different (diversity), it would be chaotic. It is the combination of these two elements that makes music beautiful to the listener.
Consider now the laws of nature. They are simple (e.g. Newton's law of universal gravitation) throughout a great diversity of objects. This makes the laws of nature beautiful.
4. The laws of nature display simplicity in diversity. (Premise)
6. Therefore, the laws of nature are beautiful. (From 2 and 4)
7. Beauty is the product of design. (Premise)
8. Therefore, the laws of nature are the product of design. (From 6 and 7)
Music isn't beautiful by chance alone, much less by some physical necessity (law). Rather, music is the expression of a personal agent, such as Mozart or Bach. What this suggests is a parallel of design between music and the laws of nature. Given that both exemplify beauty, we may infer that the laws of nature are the way they are because of the design of a personal* agent.
*Or in the view of Christians, tri-personal.