Algernon has created the "character" -- a man named Bunbury--as an excuse to get out the kinds of social engagements and obligations that bore him in the city. The story around the character is that he is in very poor health, so he needs assistance and care from his friends like Algernon. Algernon, on a moments notice can also say that he has "just heard from Bunbury who needs him immediately." This is an especially convenient story because not only can Algernon get out of whatever it is that he doesn't want to do, he can also look very altruistic while doing it; after all, he is helping out a friend in need! Who would dare to suggest that some social function like a boring society party is more important than helping a friend? It is truly a fool-proof story.
Algernon finds that the country life is more laid back than the obligations he has in town. He is perpetual debt in town, so perhaps it is difficult to live up to the expectations society would have of him. He is clearly trying to avoid filling in the dinner guest list of the society dinners hosted by his Aunt Augusta. She is the epitome of a society lady, and Algernon finds her social circle to be boring. It is ironic that he specifically mentions his distaste for a lady who "flirts endlessly with her own husband." What is funny is that he says it like that is a bad thing! But to him, it is. It is traditional, expected, and therefore, tedious. Algernon likes things to be interesting -- thus he created Bunbury. When he discovers that his friend Ernest/Jack has created a similar second identity (for himself) he calls Jack out on his "Bunburying." Jack is doing the same thing: using another person as his excuse to leave the sedate life in the country so that he can then assume that identity and live it up a bit in the more lively and interesting city. For both men, Bunburying is a relief for the obligations of their everyday lives.