Why did the anti-imperialist league believe that a policy of imperialism was inconsistent with democracy?
The American anti-Imperialist league was started in 1898 to question the annexation of the Philippines. The rationale for this is found in your question. The American anti-Imperialist league believed that imperialism was against the principles of democracy. In other words, one nation can only annex another when there is what they call "the consent of the governed." Without this consent, it would simply be a power play and would violate the will of the people.
Moreover, the league believed that if America acted in this way, they would go against their own fundamental principles, which were codified in the Declaration of Independence, more specifically, the idea of self-government. In short, America should stay out to be true to the principles of democracy and herself.
Finally, I would say that the league was heroic, because this was the heyday of imperialism. If you recall, this was the time when Europe was in Asia and Africa to colonize and take natural resources for themselves.