How does adjuvant enhance the immunogenicity of antigen?
This is a quick answer to a very broad question since the mechanisms of adjuvants are very different. I'll try to summarize just a few. For a deeper understanding please start reading this paper and its references.

1) Aluminum salts (aluminum phosphate or hydroxide)


  Studies suggest alum salts work by causing the formation of an antigen
  depot at the inoculation site from where the antigen is released slowly.
  The trapping of soluble antigen in the alum gel may also increase the
  duration of antigen interaction with the immune system. Other
  mechanisms of action involve complement, eosinophil, and macrophage,
  activation and increased efficiency of antigen uptake by antigen
  presenting cells seen with particulate matter with a size under 10
  micron.


2) Calcium salts

They work pretty much like aluminum salts but they are much better tolerated by the organism.

3) Bacteria-derived (peptidoglycan or lipopolysaccharide)

They are naturally recognized by the immune system and for this reason, they are used to simply boost the immune response.

4) Emulsions

They form a depot at the injection site, enabling the slow release of antigen. The immune system has more time to recognize the antigen and react.

5) Liposomes

They help to extend the half-life of antigens in the blood leading to a higher antigen exposure to antigen presenting cells after vaccination. A Very similar effect is achieved using microbeads made of artificial polymers.