When people are writing autobiography, what does it achieve?

I would say that depends a great deal upon the autobiography.  An autobiography can achieve something for both writer and reader, though, so let's look at these one at a time.

For the writer, an autobiography can be a form of catharsis, a way of cleansing one's mind and emotions of all that has gone before. Often people find it to be very therapeutic to share their stories with others for this reason. 

In writing an autobiography, the writer is imposing a structure upon the narrative of his or her life. This is a human need, actually, to try to make sense of the story of one's life.  Imposing that structure gives the writer a feeling of control, a way of feeling that things did not just happen, but that they have a purpose.  As we look back on our lives, we all do this. 

A writer also often has a message to share with the readers, which might be political, educational, or psychological, for example.  Many fine autobiographies have profound lessons to be shared. 

For the reader, there is often a curiosity that is satisfied in reading an autobiography. We all know of famous people we would like to know more about.  They might be actors or athletes, singers or senators.  But we want to know more, and an autobiography enables us to do so.

Some readers read autobiographies because they have a particular interest in a historical period or a particular part of the world.  An autobiography helps to round out the reader's knowledge of that world. 

We often read autobiographies because the author is in a field that we would like to succeed in. For example, someone who wishes to be a writer might read the autobiographies of writers or editors.  Someone who is interested in becoming a journalist might read autobiographies of successful reporters.  Athletes write biographies, too, that might include useful information for the person who wants to become a successful athlete.

A good autobiography is purposeful, meeting some need of writer or reader or both. I have included three links below, for more about three good autobiographies.