In the Biuret reaction, copper forms a complex with the nitrogen of the peptide bond, that looks like this (Structure 61 from the paper below):
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A strong base is necessary to provide the conditions for deprotonating these nitrogen atoms and have the conditions to form the complex. If you want to read this in detail, have a look at this publication:
* Coordinating properties of the amide bond. Stability and structure of metal ion complexes of peptides and related ligands