Why does a Classical sonata-form exposition usually move from the tonic key to the dominant key?

In Classical sonata form, the movement from the tonic (home) key to the dominant key during the exposition is fundamental to the structure's reliance on tonal contrast.12

Rather than focusing on thematic variety alone, the Classical style uses key as the primary articulator of form. By moving from the tonic to a closely related but distinct key—the dominant for major keys or the relative major for minor keys—the exposition establishes an opposition of tonal areas. This harmonic journey creates a sense of tension and conflict that the subsequent sections of the movement, the development and recapitulation, must then resolve.213

This principle evolved from earlier binary form, where the first section typically modulated to the dominant to prepare for the second section's return to the home key. In the mature Classical style of composers like Haydn and Mozart, this tonal shift became the essential framework for organizing musical drama, effectively subordinating melodic themes to their function within the broader tonal design.432