What do reviewers mean by "broad readership" when rejecting paper?

Though I haven't seen this language used, if it was the editor saying "broad leadership" rather than a reviewer, I suspect what they mean is that they did not feel your paper was sufficiently revolutionary for their tier of journal. The editor would likely base this decision on lukewarm reviews, even if those reviews are not fully negative.

Your options in this situation would be to

A) publish in a lower-tier journal, to make sure your work takes precedence in the field even if it lacks some exposure, or

B) better emphasize the novel aspects of your work in a future submission.

I also like the suggestion by @Ian_Fin that "leadership" may have been a typo from "readership" which I think would make much more sense, but my conclusions and interpretation are the same: this is the editor saying your work is not sufficiently groundbreaking for the tier of journal and either belongs in a more specialty journal or you would have needed to improve the writing to clarify the significance to the broader community.

In my experience, it means that there aren't big guys/labs/university in your list of authors/institutions. I have seen this many times, where a study on a species X from a Developing country from a big lab in a developed country gets published, and when another study on a species X from a developing country performed by scientists from the same country is not even sent to review because it is not considered interesting for the journal's readership. I am not talking here about methodological mistakes or major flaws, as sometimes those papers are not even sent for review. At the same time, I have seen papers considered uninteresting by professional editors of journals like Nature, but considered major contributions and landmark papers for journals with scientific editors (and that have open review processes). Which suggests that professional editors rely more on percieved prestige of research groups and institutions to make decissions when they do not know enough about a given subject. That is obviously bad for science.