Typically they are a class period where a teacher has a smaller group of students than normal. Exactly how it's implemented varies from school to school and teacher to teacher.
Studies have shown over and over that two of the key measures of whether a student will stay in and succeed in school through high school is whether the student has a sense of belonging at the school, and whether they have at least one adult they trust. Part of the goal of advisories is to provide students with those two things.
Responsive Advisory Meetings, if they are done properly, help students build a lot of important skills. For example, students learn how to bring up something that is bothering them to the group respectfully and productively. They learn how to advocate for themselves in terms of their education (and by extension, in other facets of their lives). They learn how to speak positively about their peers, celebrate successes, work through failures, empathize with others, and set personal goals.
Meetings also give teachers a chance to see their students in another light and vice versa. Sometimes there are games or team-building activities. Sometimes, as mentioned above, teachers can help students with their homework. (If that becomes the default activity though, it's not really a Responsive Advisory Meeting.)
When done well, they're one of the most important things schools have started doing in the last 20 years. When done poorly or lazily, or when the teacher gives the students the impression they don't care about the session, they're worthless or counterproductive. (But that's true of most classes.)