R-CO2H + R-NH2 = R-CO---NR + H2O

Does anyone know how the above reaction enables it to take part in the formation of a protein chain?

The link you show with the ---, is an amide link. Proteins are simply amino acids joined together with amide linkages.

An amino acid has a carboxylic acid group on one end and an amine on the other end.

amino acid + water(h2o)= protein