Cas13 is activated by the hybridization of a CRISPR RNA to a complementary single-stranded RNA protospacer in a target RNA. While Cas13 is not activated by double-stranded RNA in vitro, it robustly targets RNA in cellular environments where RNAs are highly structured. The mechanism by which Cas13 targets structured RNAs remains unknown. Here, we systematically probe the effects of secondary structure on Cas13. We find that secondary structure in the protospacer and 3' to it inhibits Cas13 activity and quantitatively explains the former effect through a strand displacement framework. We then harness strand displacement to generate an 'occluded' Cas13, which enhances mismatch discrimination up to 50-fold and enables sequence-agnostic mutation identification at low (<1%) allele frequencies. Using occluded Cas13, we identify human-adaptive mutations in SARS-CoV-2 and human and avian influenza A viruses, as well as oncogenic mutations in KRAS. Our work leverages improved mechanistic understanding of Cas13 to expand the scope of RNA diagnostics and enable structure-informed Cas13 approaches.