Triple-layer cuprate superconductor HgBa₂Ca₂Cu₃O₈ (Hg1223) keeps the record of the highest superconducting (SC) critical temperature T_c∼ 134K among all the existing materials at ambient pressure. T_c further increases under pressure up to T_c∼ 160K. However, its microscopic mechanism remains to be elucidated. We solve it ab initio Hamiltonians for Hg1223 using a variational solver supplemented by a neural network. The pressure dependence of the d-wave SC order parameter and estimated T_c show a T_c peak around 30GPa in quantitative agreement with the experiments. The origin of the strong SC amplitude at ambient pressure is identified as strong local Coulomb repulsion U attributed to poor screening. Further increase in T_c under pressure is understood from interplay of three elements, namely increased electron hopping t, decreased U and more importantly, strongly reduced offsite Coulomb repulsion V with increasing pressure. Pairing mechanism is identified as the emergent local attraction counterintuitively generated from the originally strong local repulsion U. The emergent attraction is interpreted from ``attraction from reduced repulsion'', originating from the release of the fluctuating doubly-occupied sites characterized from the ``false vacuum'' in the Mott insulator to the double-occupation-free d-wave SC states upon carrier doping. This instantaneous attraction is in contrast with the conventional BCS SC mediated by bosonic glues. The local attraction is consistent with the electron fractionalization supported in experimental analyses. The coexistence of the SC and antiferromagnetic order is also demonstrated as a characteristic feature of the multilayer system. The microscopic understanding of Hg1223 offers a new route explicitly using this emergent attraction to design and optimize SC materials.