Governments need to know who is inside their borders for welfare and policing. They need to know who visits and who wishes to emigrate. The borders lets a government know if a person should be paying tax in their country or not.
A country is really just a group thinking for the people inside. This itself has led to sections of countries breaking away. Take Pakistan for instance. Originally part of India but a large population of Muslims who felt that they did not have equal standing in a nation with Hindu majority wanted their own independence. Eventually Pakistan was formed with outside help and a new country was born.