How do people become homelessness?

According to the National Law Center on Homelessness and Poverty, insufficient income and lack of affordable housing are the leading causes of homelessness in the United States.
Major factors that contribute to a lack of sufficient income to afford housing include mental illness and substance addiction. That is, mental illness and substance addiction can destabilize an individual's economic life to the point that they are unable to earn any, or enough, income to afford housing.
Finally, among homeless youth, a lack of familial support structures and absence of basic life skills knowledge created as a result of insufficient foster care resources is a major precipitating factor to homelessness.
Factors contributing to homelessness outside the United States may, depending on geography, differ significantly from those contributing to homelessness within the United States. In some areas, war and natural disaster can be a leading cause of homelessness.
http://web.stanford.edu/class/e297c/poverty_prejudice/soc_sec/hcauses.htm