The term "attentive public" comes from the part of the field of political science that is concerned with public opinion. This subfield studies the way that public opinion is formed and its impact on the political process.
The "attentive public" for any given political issue is that segment of the public that pays attention to news about that issue and cares about what happens with that issue. This means that there are different "attentive publics," for example, for issues like gay marriage on the one hand and the level of tariffs on the other.
The concept of the "attentive public" is important because the opinions of these people matter more (and are often more worthy of study) than the opinions of the public as a whole since the public as a whole may not care much about a given issue.