We report an in-plane optical spectroscopy study on the iron-selenide superconductor K$_{0.75}$Fe$_{1.75}$Se$_2$. The measurement revealed the development of a sharp reflectance edge below T$_c$ at frequency much smaller than the superconducting energy gap on a relatively incoherent electronic background, a phenomenon which was not seen in any other Fe-based superconductors so far investigated. Furthermore, the feature could be noticeably suppressed and shifted to lower frequency by a moderate magnetic field. Our analysis indicates that this edge structure arises from the development of a Josephson-coupling plasmon in the superconducting condensate. Together with the transmission electron microscopy analysis, our study yields compelling evidence for the presence of nanoscale phase separation between superconductivity and magnetism. The results also enable us to understand various seemingly controversial experimental data probed from different techniques.