Epitranscriptomic RNA modifications can regulate mRNA function; however, there is a major gap in our understanding of the biochemical mechanisms mediating their effects. Here, we develop a chemical proteomics approach relying upon photo-cross-linking with synthetic diazirine-containing RNA probes and quantitative proteomics to profile RNA-protein interactions regulated by N-methyladenosine (mA), the most abundant internal modification in eukaryotic RNA. In addition to identifying YTH domain-containing proteins and ALKBH5, known interactors of this modification, we find that FMR1 and LRPPRC, two proteins associated with human disease, "read" this modification. Surprisingly, we also find that mA disrupts RNA binding by the stress granule proteins G3BP1/2, USP10, CAPRIN1, and RBM42. Our work provides a general strategy for interrogating the interactome of RNA modifications and reveals the biochemical mechanisms underlying mA function in the cell.