Details

CopyFunction takes a function that produces a box expression.
TemplateBox parameters are substituted into the function to create the box expression that is sent to the kernel when the box is evaluated.
When a TemplateBox is copied, a full-fidelity version is placed on the clipboard. If the clipboard is then used to paste into an external text editor, CopyFunction controls how the TemplateBox is converted to plain text.
CopyFunctionAutomatic causes the DisplayFunction to be used if the TemplateBox is copied as plain text. If the TemplateBox is copied as input text, the InterpretationFunction will be used.

Examples

Set a copy form different from the displayed or interpreted form:

Wolfram Language code: RawBoxes[TemplateBox[{}, "template", DisplayFunction -> ("fish"&), CopyFunction -> ("fowl"&)]]

Here is the pasted form:

fowl

Create a copy form from TemplateBox arguments:

Wolfram Language code:

RawBoxes[TemplateBox[{"f", "x"}, "template", DisplayFunction -> (RowBox[{#1, " of ", #2}]&), CopyFunction -> (RowBox[{#1, "[", #2, "]"}]&)]]

Here is the pasted form:

f[x]

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