Organic-inorganic nanocomposites based on heat-resistant crosslinked polycyanurate (PCN) and N-phenylaminopropyl polyhedral oligomeric silsesquioxane (NPAP-POSS), containing eight reactive secondary amino groups, were synthesized using the in situ reactive formation method. Fourier transmission infrared (FTIR) spectroscopy and dynamic differential scanning calorimetry (DSC) methods were used to study the effect of NPAP-POSS on the kinetics of bisphenol E dicyanate ester (DCBE) polycyclotrimerization during the formation of PCN in PCN/NPAP-POSS nanocomposites. The content of the nanofiller was varied from 0.05 to 1.00 wt.%. Based on the results of FTIR spectral studies, the main kinetic peculiarities of PCN formation were found and their changes under the action of NPAP-POSS nanofiller were determined. A significant catalytic effect of NPAP-POSS on the polycyclotrimerization of DCBE was found, which is confirmed by a decrease in the time of the onset of auto-acceleration, an acceleration of the conversion of cyanate groups of DCBE and the formation of triazine cycles of PCN, an increase in the values of the maximum reaction rate, a decrease in the duration of the reaction, etc. The dynamic DSC method also confirmed the catalytic effect of NPAP-POSS on the formation of PCN in the nanocomposites and established the main kinetic characteristics depending on the content of the nanofiller: a significant decrease in the temperature of the exothermic maximum, an increase in the reaction enthalpy, non-monotonic changes in the induction period and reaction rate, etc. From the analysis of the experimental data, it was concluded that the detected changes in the kinetics of the in situ reaction formation of PCN/NPAP-POSS nanocomposites and the recorded catalytic effect of the nanofiller are due to the fact that two chemical processes occur during the synthesis of the nanocomposites: chemical interaction of –O–C≡N groups of DCBE with secondary –NH groups of NPAP-POSS, which led to further embedding of nanoparticles into the resulting polymer matrix and the direct polycyclotrimerization of DCBE with formation of hybrid polycyanurate network. Schemes of the sequential reactions explaining the catalytic effect of the nanofiller in the synthesis of hybrid PCN/NPAP-POSS nanocomposites are proposed. It was concluded that under the selected conditions of the synthesis, the greatest catalytic effect of the nanofiller is manifested at its content of 0.10 wt.%, since for this sample the maximum shift of the reaction exothermic peak towards lower temperatures, the maximum reaction rate, and the minimum induction period and reaction start temperature were recorded. The results of the research make it possible to optimize the synthesis of heat-resistant materials promising for use in special-purpose structures.