Microsoft Azure Stack primarly focus on customer’s whose workload is difficult to be brought to Azure Cloud (Public Cloud) due to various reasons. These reasons can be due to legal, business functions or applications limitaitons. Eg: Banks are restricted to reatain their data within the country itself…etc
Being said that Azure Stack is a pre-build (Customized) rack coming pre-configured everything for your company. Starting from minimum 4 servers all the way up-to 16 nodes. Just to mimic Azure Public cloud in future customers can have multiple racks of Azure Stack in their datacenter.
Servcie providers use Azure Stack to provide multi-tenant model business offerings to their customers.
Azure Stack will connect with Microsoft Azure when it coming to billing the actual usage of the resources.

Microsoft’s release of Azure Stack, an on-premises version of its public cloud, could be important for networking and data center pros for one simple reason: It gives customers a way to use a popular and familiar cloud platform without shipping their sensitive data into a multi-tenant environment.

Azure Stack is software from Microsoft that’s been certified to run on a select group of partners’ hardware and is intended to look and feel just like the Azure public cloud. In addition to providing a common management platform between the public and private cloud, Azure Stack is important for another reason too: none of Microsoft’s biggest public cloud competitors have anything like it.