Why does a real world capacitor behave like an inductor at frequencies above its self-resonant frequency?
The behavior of a realistic multi-layer ceramic capacitor is determined by its construction. It is made of ceramic pieces with conducting surfaces, which are connected together at collector electrodes. 

Unfortunately, every conductor posess some self-inductance, which begins to play dominant role at higher frequencies. One good article about equivalent models of MLCC is presented by Taiyo Yuden corporation, as illustrated below:

 

For larger-size (more layers) the equivalent circuit grows up (see the article), so the effective characteristics change accordingly.