This joke plays on the word "halálom", which means "my death". "Halálom" is by chance made up of the words "hal" - "fish" and "álom" which is "dream". So "Fishdream" is the same as "My death". The joke is: "Which animal is it scary to dream of? A fish. Why? It's about my death / fishdream."
Otolaryngologist (An ear and throat specialist is "gégészet" in Hungarian, while the sound geese make is "gá gá" in Hungarian, so "gégészet" becomes "gágászat")
"Hajbókolás" means giving compliments to someone (like royalty), but in this form "haj" also means "hair", and "bókolás" means "paying compliments to someone". So the man is complimenting the hair.