The Archive attribute was first introduced by MS-DOS version 2.0 when the XCOPY utility was also created. Therefore, the Archive attribute is probably best explained by how XCOPY handles the Archive attribute. The purpose of the Archive atribute was clearly to quickly determine whether a file requires a back up (archiving).
The Archive attribute is set whenever an existing file is either overwritten or modified (i.e., renamed or moved to another directory) by the file system. A new file is usually created with the Archive attribute set.
The idea is to have a utility or application program to take the responsibility of clearing the Archive bit when a file is successfully backed up.
The next system backup operation will be made much more efficient by focusing only on the files with the Archive attribute bit set which are either newly created or modified in any way since all the Archive attributes are cleared the last time (i.e., the last backup time).
