In an API-First approach APIs are treated as “first-class citizens”, that expose all business functionality thru a well-defined set of contracts that allows two or more systems to interact. 
API-First is closely associated to Microservices; the defined contracts are the entry point to all the service business functionality.
Having these contracts in place, that oftentimes follow API description languages such as Open API, contribute to a robust design and foster collaboration between different technical and non-technical stakeholders.
This strategy allows businesses to create APIs that can be efficiently developed for all types of devices and platforms and can serve all types of applications in a multi-channel environment.