The significance of API gateways in microservices architecture lies in their role as a single entry point for client requests to multiple backend services. They streamline communication, manage traffic, enforce security policies, and provide functionality like load balancing and request routing. By abstracting the underlying microservices, API gateways simplify the client-side architecture and improve performance. They also facilitate service discovery and enable monitoring and analytics, which are essential for managing distributed systems effectively and ensuring a seamless user experience.