Polyimides (PIs) have attracted great attention because of their excellent properties and applications in areas such as flexible display substrates, microelectronics, and integrated circuits. The effects of crosslinking on the thermal, mechanical, and dielectric properties of PIs, however, still require further investigation. In this work, we introduce a crosslinkable monomer, bicyclo[2.2.2]oct-7-ene-2,3,5,6-tetracarboxylic acid dianhydride (BTA), which contains double bonds, into the system consisting of cyclobutane-1,2,3,4-tetracarboxylic dianhydride (CBDA) and 4,4′-oxydianiline (ODA) for the synthesis of PI films. The obtained PIs are further crosslinked by free radical reactions using a crosslinking agent, 1,3,5-triallyl-1,3,5-triazine-2,4,6(1H,3H,5H)-trione (TAIC) to form three crosslinked PI materials, PI-TAIC-5%, PI-TAIC-10%, and PI-TAIC-15%. The resulting all three crosslinked PI films exhibit excellent thermal properties with their glass transition temperatures (Tg) higher than 400 °C. For the mechanical properties, the formation of the rigid network structure leads to an improvement in their tensile strengths and Young's moduli. The introduction of 10 mol% TAIC in the PI (PI-TAIC-10%) shows the highest tensile strength (114.77 MPa), representing an 18.6% enhancement in comparison with that of the original PI material. In terms of dielectric properties, the crosslinking reactions effectively increase the free volumes between the polymer chains and reduce the molecular dipole moments. As a result, PI-TAIC-15% significantly reduces the dielectric properties, resulting in a decrease in the dielectric constant (Dk, from 3.42 to 3.15) and the dielectric loss (Df, from 3.67 × 10−2 to 2.58 × 10−2) under 10 GHz frequency. As far as the coefficient of thermal expansion (CTE) results, the formation of the network structure in PI-TAIC-15% restricts the movement of the polymer chains, leading to a 15% reduction in the CTE value than that of the non-crosslinked PI film. Overall, this work provides a promising strategy to optimize the mechanical and dielectric properties of high thermally stable PIs via crosslinking reactions.