In addition to moiré superlattices, twisting can also generate moiré magnetic exchange interactions (MMEIs) in van der Waals magnets. However, owing to the extreme complexity and twist-angle-dependent sensitivity, all existing models fail to fully capture MMEIs and thus cannot provide an understanding of MMEI-induced physics. Here, we develop a microscopic moiré spin Hamiltonian that enables the effective description of MMEIs via a sliding-mapping approach in twisted magnets, as demonstrated in twisted bilayer CrI. We show that the emergence of MMEIs can create a magnetic skyrmion bubble with non-conserved helicity, a 'moiré-type skyrmion bubble'. This represents a unique spin texture solely generated by MMEIs and ready to be detected under the current experimental conditions. Importantly, the size and population of skyrmion bubbles can be finely controlled by twist angle, a key step for skyrmion-based information storage. Furthermore, we reveal that MMEIs can be effectively manipulated by substrate-induced interfacial Dzyaloshinskii-Moriya interactions, modulating the twist-angle-dependent magnetic phase diagram, which solves outstanding disagreements between theories and experiments.