The Vonage Voice API allows developers to play audio streams into calls, providing various use cases such as playing caller on-hold music, conference call scenarios, and pre-recorded messages, by utilizing the Call Control Object (NCCO) or Voice API (VAPI). The NCCO provides a convenient way to control an inbound call, while the VAPI offers more flexibility in streaming audio into conferences. Developers can implement these features using Vonage's starter applications and code samples, which provide a step-by-step guide on deploying the apps and setting up the necessary credentials.