There is no single accepted explanation of the term "Chicano."
My understanding of the origins of this term is taken from the Encyclopedia of Race and Racism, published by MacMillan Reference USA. In the entry for the term "Aztlan," the encyclopedia argues that the term "Chicano" comes from the word "Mexicano."
It is a fact that the people that we refer to as "Aztecs" called themselves "Mexica." This word was pronounced with the "x" having a "sh" sound. This meant that the word "Mexicano" could be pronounced "Meshicano." From there, the sound shift to "Mechicano" is a small one, and the term "Chicano" can easily be seen as an abbreviation of this word.
This is, of course, simply an informed supposition on the part of scholars. There is no direct proof of the origin of the word.