Although citations are widely used to measure the influence of scientific works, research shows that many citations serve rhetorical functions and reflect little-to-no influence on the citing authors. If highly cited papers disproportionately attract rhetorical citations then their citation counts may reflect rhetorical usefulness more than influence. Alternatively, researchers may perceive highly cited papers to be of higher quality and invest more effort into reading them, leading to disproportionately substantive citations. We test these arguments using data on 17,154 randomly sampled citations collected via surveys from 9,380 corresponding authors in 15 fields. We find that most citations (54%) had little-to-no influence on the citing authors. However, citations to the most highly cited papers were 2–3 times more likely to denote substantial influence. Experimental and correlational data show a key mechanism: displaying low citation counts lowers perceptions of a paper's quality, and papers with poor perceived quality are read more superficially. The results suggest that higher citation counts lead to more meaningful engagement from readers and, consequently, the most highly cited papers influence the research frontier much more than their raw citation counts imply.