Why would someone have signed, approved, or supported the Declaration of Independence during this time

Supporters and signers of the Declaration of Independence believed firmly that each person was put on Earth by a creator with natural rights and that no King across a sea could restrict those rights.

Believing that you have rights and being told by a person thousands of miles away that you don't is a pretty powerful motivator. So, supporters were willing to die for the belief that they had rights that could not be restricted.