AES is deemed secure because:
- Its building blocks and design principles are fully specified.
- It was selected as part of an open competition.
- It has sustained 15 years of attempted cryptanalysis from many smart people, in a high-exposure situation, and it came out relatively unscathed.
Another reason, which is not as good but felt important by many people:
- It was designed by non-American cryptographers.
In asymmetric cryptography we often (try to) reduce security to a "known hard problem", a luxury which is not often encountered in symmetric cryptography, but this does not change the conceptual root of the issue: at some level, there is some "problem" for which no efficient solving algorithm is known, despite decades of research. It is not proven that the problem is necessarily hard, or even that there can exist such as thing as a necessarily hard problem, but we are just stumped when it comes to finding a solving algorithm. With AES, the "hard problem" happens to be the AES itself.