To smooth, shape, or wear away something with a file or filing tool.
From 'be-' prefix combined with 'file', the tool. This represents an English strategy of converting nouns into verbs using the 'be-' prefix, especially when the noun represents an implement that acts upon something.
In the heyday of the 'be-' prefix (roughly 1200-1700), English speakers could take almost any tool name and create a verb—'befinger' meant to handle with fingers, 'beclothe' meant to dress, and 'befile' meant to work something with a file, showing how productive language creation was before we standardized our vocabulary.
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