This is a phrase (“on [purely] objective grounds”) I have used rather frequently in the notes and articles posted here. The purpose of the phrase is to indicate when a saying, narrative, or other tradition recorded in Scripture may be considered as authentic on the grounds of critical scholarship, without resort to any doctrine regarding the inspiration or historical reliability of Scripture. Similarly, it is used to judge the greater likelihood of various (critical) theories related to the development of tradition and how the Scriptures (the Gospel narratives, especially) came to be composed. For more traditional-conservative commentators, and for many devout believers in general, the accuracy and authenticity of the Scriptures is self-evident—is assumed or taken for granted—and requires no (objective) critical analysis to confirm the matter. However, even for those who hold, or tend toward, the traditional-conservative position, the observations and insights of critical scholarship can be most beneficial: it is foolhardy (in the worst sense) to ignore or disparage them, and, I should say, unworthy of the believer who wishes to be a serious student of the Scriptures.
The qualifying term “objective” implies verifiable evidence, both internal and external to Scripture, which can be analyzed, agreed upon, and accepted, by all commentators—believer and non-believer alike—apart from what one personally believes or thinks about the Scripture. This is contrasted with (or, one may say, complemented by) interpretation on “subjective” grounds—that is, the personal (whether unique or shared by a wider community) opinion or belief of the commentator. Examples of “objective” evidence include: word usage, the development and particular meaning of a word or phrase, historical parallels to a word or passage, similarities of usage in other writings, signs of historical/literary development in a narrative, and so forth. Complete objectivity may or may not be possible for a scholar or commentator, but it remains a noble goal, and one which ought to be pursued in faith and humility.