AC TO DC Converter IC
Regarding the usage circuit of the LNK586, you find it on page 4 of the datasheet and I copied it here:



The blue part is a rectifier (AC to DC), the orange part is the IC which forms the control part of a DC/DC convertor (green part)

Regarding your second question: You can find many ICs that make the same thing (control a DC/DC convertor), however, to design an offline (direct from AC mains) power supply is not trivial at all. To get a circuit that is significantly more compact that the module you’re currently using would be extremely challenging: most of the space in your current solution is most likely used by the passive components, to reduce them you would need to increase the switching frequency of the DC/DC convertor but that would increase the switching losses in the power transistors and magnetic components at the same time.