In idiomatic speech, to "kick off" something is to begin something; that is, to be the first to do something.
Example: In a recent "The Early Show" segment on CBS, two political commentators were being consulted by a CBS hostess about recent political events: (Republican) Ann Coulter, and (Democratic) Tanya Acker. The first question concerned credit for the U.S. withdrawal of combat troops from Iraq for President Obama. The second question concerned what the Associated Press now calls the "NY Mosque" controversy.
Ann Coulter had answered the segment's first question before Tanya. When the hostess posed the second question, she said, "Tanya, I'll let you kick this one off." This meant, Tanya would be permitted to answer first for the second part of the segment. It is in this sense that she was "kicking off" the second part.
(English Idioms takes no political positions, but this is a real-life example of the idiom.)