A "thick description" is the intellectual and literary act of describing what happens during an interview with a research participant. It's a reflection upon what one saw as an interpretation of behavior within a certain context. Or as Geertz says, it's a sorting through of "webs of significance that (man) himself has spun."
When the Internet caught on over a decade ago, companies wanted to learn why people were browsing and what exactly they were doing. So, design researchers popularized a method called contextual inquiry, which is the term given to the close study of what happens between a person, their computer, and the setting they're in. It was (and remains) a quick way for designers to focus on observing people doing specific tasks in order to learn how they do them, and, to some extent, why they do them. But contextual inquiry, while appropriate for some design projects, doesn't really enable you to discover the more nuanced aspects of people's thoughts and behaviors and their relationship to a broader culture—all pretty important things to unearth if you're tasked with innovating a product or service. A thick description does by expanding the designer's field of vision to include things on the periphery of the participant's life that, while indirect to the topic of study, influence them in meaningful ways.
One of the keys to this research method is how well we, as researchers, talk to and observe people when we're in the field—or rather, the depth to which we are able to go during these talks and observations. What did we see and hear? What was significant? What was peculiar or distinct, and how does it relate to the larger set of behaviors observed?