Gardner's work on multiple intelligences is so important because it suggests that we ought to be thinking about intelligence and teaching in a different way than we have been.
Generally, our schools have valued the kind of people that have what Gardner calls "linguistic intelligence" and those who have what he calls "logical-mathematical intelligence." We have thought that these people are the only ones who are "intelligent" and that people who do not show these kinds of intelligence simply aren't smart. We also teach in ways that rely on those kinds of intelligence.
If we take Gardner's work seriously, we have to realize that there are these other kinds of intelligence and that it is important to value the abilities of those who have them. We also have to try, as teachers, to figure out ways to teach our subjects through the use of other kinds of intelligences so that we can reach a broader range of students.