With a serverless architecture the client manages the software while the service provider, such as AWS, Microsoft Azure or Google Cloud, manages everything else.
Applications still run on a server but a provider manages that server. By allowing the service provider to manage and maintain the infrastructure more resources can be focused on developing the product.
Providers automatically scale applications horizontally allowing for multiple instances of a function to run concurrently without any impact on performance.
The microservices that applications depend on are also hosted in the cloud allowing for more efficient testing and faster implementation of new features.