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Context. The Spitzer Space Telescope has detected a powerful ($L_{\rm H_2}\sim10^{41}$ erg s-1) mid-infrared H2 emission towards the galaxy-wide collision in the Stephan's Quintet (henceforth SQ) galaxy group. This discovery was followed by the detection of more distant H2-luminous extragalactic sources, with almost no spectroscopic signatures of star formation. These observations place molecular gas in a new context where one has to describe its role as a cooling agent of energetic phases of galaxy evolution. Aims. The SQ postshock medium is observed to be multiphase, with H2 gas coexisting with a hot (${\sim}5$ $\times$ 106 K), X-ray emitting plasma. The surface brightness of H2 lines exceeds that of the X-rays and the 0-0 S(1) $\rm H_2$ linewidth is ${\sim}900$ km s-1, of the order of the collision velocity. These observations raise three questions we propose to answer: (i) why is H2 present in the postshock gas? (ii) How can we account for the $\rm H_2$ excitation? (iii) Why is H2 a dominant coolant? Methods. We consider the collision of two flows of multiphase dusty gas. Our model quantifies the gas cooling, dust destruction, H2 formation and excitation in the postshock medium. Results. (i) The shock velocity, the post-shock temperature and the gas cooling timescale depend on the preshock gas density. The collision velocity is the shock velocity in the low density volume-filling intercloud gas. This produces a ${\sim}5$ $\times$ $10^6~$K, dust-free, X-ray emitting plasma. The shock velocity is lower in clouds. We show that gas heated to temperatures of less than $10^6~$K cools, keeps its dust content and becomes H2 within the SQ collision age (${\sim}5$ $\times$ $10^6~$years). (ii) Since the bulk kinetic energy of the H2 gas is the dominant energy reservoir, we consider that the H2 emission is powered by the dissipation of kinetic turbulent energy. We model this dissipation with non-dissociative MHD shocks and show that the H2 excitation can be reproduced by a combination of low velocities shocks ($5{-}20$ km s-1) within dense ($n_{\rm H}>10^3~{\rm cm}^{-3}$) H2 gas. (iii) An efficient transfer of the bulk kinetic energy to turbulent motion of much lower velocities within molecular gas is required to make H2 a dominant coolant of the postshock gas. We argue that this transfer is mediated by the dynamic interaction between gas phases and the thermal instability of the cooling gas. We quantify the mass and energy cycling between gas phases required to balance the dissipation of energy through the H2 emission lines. Conclusions. This study provides a physical framework to interpret H2 emission from H2-luminous galaxies. It highlights the role that H2 formation and cooling play in dissipating mechanical energy released in galaxy collisions. This physical framework is of general relevance for the interpretation of observational signatures, in particular H2 emission, of mechanical energy dissipation in multiphase gas