Full Idea
Frege's contribution with respect to the definition of equinumerosity was to replace Cantor's sets as the objects of number attributions by concepts.
Gist of Idea
Frege replaced Cantor's sets as the objects of equinumerosity attributions with concepts
Source
report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by William W. Tait - Frege versus Cantor and Dedekind XII
Book Reference
'Philosophy of Mathematics: anthology', ed/tr. Jacquette,Dale [Blackwell 2002], p.59
A Reaction
This pinpoints Frege's big idea, which is a powerful one, and may be right. The difficulty seems to be that the extension is ultimately what counts (because that is where plurality resides), and it is tricky getting the concept to determine the extension.