Websites can have two sides. The side that the user interacts with is called the “front end”. The side that deals with logic and data is called the “back end”.
A developer that only deals with the way things look (typography, layout, color schemes etc.) is called a “Front end developer”. A developer that only deals with the way things work under the hood (logic, servers, security etc.) is called a “back end developer”. Finally, a developer that is responsible for the implementation of both sides is called a “full stack developer”.
Full stack development encompasses the planning, design, development, deployment, and maintenance of a website/application.
A developer that has the skillset to do a wider range of work is more productive than one that does not. They also save time and money. Think - you have a front end developer that knows nothing about server side logic. Now you need to find a back end developer to work alongside your front end developer. What if they don't get along? What if they have incompatible approaches to problem solving? What if the new back end developer creates a mess that your front end developer has no idea how to address?
A full stack developer can handle everything without all the extra fluff. You tell a fullstack developer what to do, they'll leverage their skillset to get the job done. Full stack developers are deeply committed to their craft and can be a force multiplier to any organization's IT systems.