In order to examine the psychology of satire, one needs to understand both satire and how satire has been used throughout history.
Understanding satire is by far the easiest of the two. According to the eNotes site, satire is defined as
the use of humor and wit with a critical attitude, irony, sarcasm, or ridicule for exposing or denouncing the frailties and faults of mankind’s activities and institutions, such as folly, stupidity, or vice. This usually involves both moral judgment and a desire to help improve a custom, belief, or tradition.
The harder part of understanding the psychology of satire involves research into the works and minds of those who wrote incorporating satire into their texts. Historically, the most famous satirical authors were Pope (example of satirical text "The Rape of the Lock"), Dryden (in "To His Sacred Majesty"), Kingsley Amis (in "Laughter's To Be Taken Seriously"), and Chaucer ("The Canterbury Tales").
Each of these authors wished to show the problems associated with the world as society and governments ruled it. Each had a problem with the government, the Church, or the nobility which decided what was proper and to be expected/accepted. Through their texts, the authors showed the problems with the mentalities which they depicted in the world around them. Many times, the text forced people to look at society around them, new texts were written to support the new and novel ideas, and (most importantly) people began to question life as it was.
What this did was bring new ideologies and movements into both the literary world and the world at large. For example, Dryden's work was so influential that a new period was established: The Age of Dryden.