"Independence" means being able to do things on your own without needing help or permission from others. It is about having the freedom to make your own decisions and take care of yourself without relying on someone else. Full definition
The line is a declaration of independence of sorts. (4columns.org)
Presumably he's concerned because many people who consider themselves «independent» (party registration «blank») believe the «independence party» represents their registration. (nystateofpolitics.com)
Alternatively, you can also decide the time required to reach your financial independence day and discover the saving rate you will use as a basis for creating a budget. (debtfirms.com)