The recently discovered kagome superconductors AV_{3}Sb_{5} exhibit tantalizing high-pressure phase diagrams, in which a new domelike superconducting phase emerges under moderate pressure. However, its origin is as yet unknown. Here, we carried out the high-pressure electrical measurements up to 150 GPa, together with the high-pressure x-ray diffraction measurements and first-principles calculations on CsV_{3}Sb_{5}. We find the new superconducting phase to be rather robust and inherently linked to the interlayer Sb2-Sb2 interactions. The formation of Sb2-Sb2 bonds at high pressure tunes the system from two-dimensional to three-dimensional and pushes the p_{z} orbital of Sb2 upward across the Fermi level, resulting in enhanced density of states and increase of T_{C}. Our work demonstrates that the dimensional crossover at high pressure can induce a topological phase transition and is related to the abnormal high-pressure T_{C} evolution. Our findings should apply for other layered materials.