The quick answer is that it refers to the software products that a marketing team uses to execute their digital campaigns. These systems should communicate with each other, so data flows from one product to the next. Prospects and customers generally won’t know what mix of products a company is using; these tools all work behind the scenes.
It’s a relatively new buzzword that marketers stole from the IT side of the house. You’d often hear what Technology Stack a company was using (e.g., “They are using LAMP. Linux as the operating system, Apache as the Web server, MySQL as the relational database management system, and PHP for their scripting language.”). A developer that could work on all aspects of their technology was labeled a Full-Stack Developer, and was a hot-commodity. Of course, marketers grabbed that term too, and now someone that understands all the components of a digital marketing program can claim the title of a Full-Stack Marketer.