The main object studied in harmonic analysis are locally compact (Abelian) groups. As several branches of number theory study locally compact fields, which are, in particular, locally compact groups, all the results on locally compact groups known in harmonic analysis are applicable in the study of locally compact fields.
A classical text on number theory that utilises harmonic analysis is André Weil's Basic Number Theory. The first chapter of this book is devoted to locally compact fields and utilises several results of harmonic analysis such as the existence and uniqueness of the Haar measure. It might be instructive to have a look into the book by Weil in order to see why harmonic analysis is a rather natural tool for studying locally compact fields.
Another, more modern, text that discusses similar topics as Weil's book is Fourier Analysis on Number Fields written by D. Ramakrishnan and R. J. Valenza. This book develops all the necessary theory of harmonic analysis in the first three chapters. This development clearly shows which theory of harmonic analysis is useful in number theory and which not. As the title Fourier Analysis on Number Fields indicates, it is related to Tate's thesis, which was called Fourier Analysis in Number Fields, and Hecke's Zeta functions. The remaining chapters of the book deals with topics introduced in Tate's thesis.