Background
Customer Success is closely related to the maturation of the SaaS (Software as a Service) model, famously leveraged by the giant B2B software Salesforce.
Software used to be a synonym for complexity and bureaucracy. It used to be normal to deploy multi-month and multi-million projects on-premises, making it difficult for customers to leave.
Salesforce decided to spin the power balance on its head, making it easy for customers to buy and cancel software, thanks to its business model based on the cloud.
Even with this evolution and consequent sales success, Salesforce realized churn was eating through its growth and devised a strategy that came to be known as Customer Success.
What is Customer Success?
Customer Success aims at putting the customer at the center of every interaction between the customer and your company.
Consequently, it helps customers attain their goals by interacting with your team, product, and services.
That's why it's paramount to keep track of metrics such as churn, NPS, CSAT, MRR, and LTV.
They help identify gaps between your company's strategy executions and customer's expectations, thus feeding improvements back to all areas.
Customer Success vs. Customer Support
Customer Support focuses on responding to customer enquires and issues. It is a reactive approach by design: customers raise a request, and the support team addresses it.
On the other hand, Customer Success anticipates customer needs and improvements. It is achieved by monitoring key indicators continuously and having frequent touchpoints with customers.
Customer Success vs. Sales
Although renewals, upselling and cross-selling can be part of the Customer Success activities, its scope of work is much broader. 
It encompasses indicators about operations, usage, satisfaction, and process compliance. 
Sales activities are a consequence of all the actions performed by the Customer Success team.