How did all these people contribute to the analysis of literature for children?

Vladimir Propp, Bruno Bettelheim, Maria Tatar and Kieran Egan were who  studied a specific area or psychological aspects of kids' mind to know in details their reactions, thoughts as well as feelings while reading fairy tales. 

Vladimir Propp:   
He was born on April 17th in St. Petersburg in 1895, Vladimir developed a Russian Formalist Approach to study and analyse fairy tales. Propp also argued that all fairy tales were constructed of certain plot elements, which he called "functions" that occurred in an determined sequence. These "functions" were reconnaissance, delivery, trickery, complicity, villainy, mediation and the different roles of heroes and villains could play in a story. Each function explained how they worked jointly, however, fairy tales also needed several other elements such as their respective plot, settings, tone, and characters in order to gain a cohesive and well-written tale for children.

Bruno Bettelheim: 
He was born on August 28th in Viena in 1903. He was a controversial Austrian Psychoanalyst and during his studies in Viena, Bruno was influenced by World War I, the Bolshevik Revolution, and Sigmund Freud.
In the United States,  he worked as a Psychologist and Psychiatric Professor at the University of Chicago, he viewed children's behaviour as resulting from overcame negative parents interacting with them during critical early stages in their Psychological development. He wrote a book called "The Uses of Enchantment", which had a popular Psychoanalytical look at fairy tales. Bruno contributed that classic fairy tales had an unique importance and impact on children's development, which could be explored by studying their styles, characters and themes.
Styles were related to happy endings in which good was rewarded and evil was punished. Bruno mentioned that happy endings satisfied children's spiritual desires.
Characters seemed to be a significant element in every fairy tale. They could be heroes or heroines . According to Bruno Bettelheim, children tried to imitate their manners as well as their qualities to be successful in life. He denied that the evil played  negative effects, because kids learnt that they must not do bad things to appeal punishments and they also learnt how to deal successfully with their fears.
The theme also played a primordial role in fairy tales. From Bruno's point of view, the theme depended on what the story was about and it transmitted a meaningful message for kids to develop independence for the future.

Maria Tatar:

 Tatar was interested in how the fairy tales were first written down, the ways in which the texts reflected the historical realities of another time and place and the Psychological effects.  Maria showed, these tales helped children to survive in the world ruled by adults. Maria also believed that fairy tales were connected with all kind of adult secrets for they told children about death, romance, marriage and, in some cases, they would speak about sex and violence. As regards violence, Fairy tales were often violent but they acted as a therapy  for kids. Maria Tatar added that  violence helped little ones to face their fears, for which they did not yet the exact language developed. 
Maria Tatar expressed that stories shared moral aspects, giving life's lessons and transmitting wonderful messages for kids. Nevertheless, she explained that moral was added to fairy tales when they were rewritten for children.

Kieran Egan:
He was born in Ireland in 1942. He found out Education involved some kind of development on children, especially intellectual tools, like Language for example. 
Egan developed an Imaginative Approach to Teaching to help kids to become more knowledgeable and more creative in their thinking. In fact, that new approach offered an understanding of how children's imagination worked in learning and showed how children acquired cognitive tools to promote creativity in the classroom.  Kieran proposed the use of intellectual tools, particularly Language or Literacy to generate successive kinds of understanding  like somatic, mythic, romantic, philosophical, and ironic. Egan concluded that if teachers put into practice them, their way of teaching would change completely and children would be more engaged in the classroom.

Eventually, Egan focused on Teaching Core Literacy Skills,  kids internalized and retained material to interpret through their imagination and emotions pieces of reading and writing.

To put it briefly, each of these well-known professionals contributed through different techniques to the analysis of literature for children.

References: 

Maria Tatar. [ONLINE] Available at: http://people.fas.harvard.edu/~tatar/Maria_Tatar/About_Me.html. [Last Accessed 26 October 2012].
Ana moreno (1997). El caso de Bruno Bettelheim. [ONLINE] Available at: http://www.revistacriterio.com.ar/sociedad/el-caso-de-bruno-bettelheim/. [Last Accessed 26 October 2012].

Molly Finn (1997). In the Case of Bruno Bettelheim . [ONLINE] Available at: e.g. http://www.microsoft.com. [Last Accessed 26 October 2012].


(2001). . [ONLINE] Available at: http://www.sfu.ca/~egan/. [Last Accessed 26 October 2012].