The Multi-slit Solar Explorer (MUSE) is a proposed mission composed of a multislit extreme ultraviolet (EUV) spectrograph (in three spectral bands around 171 Å, 284 Å, and 108 Å) and an EUV context imager (in two passbands around 195 Å and 304 Å). MUSE will provide unprecedented spectral and imaging diagnostics of the solar corona at high spatial (≤0.″5) and temporal resolution (down to ~0.5 s for sit-and-stare observations), thanks to its innovative multislit design. By obtaining spectra in four bright EUV lines (Fe IX 171 Å, Fe XV 284 Å, Fe XIX-Fe XXI 108 Å) covering a wide range of transition regions and coronal temperatures along 37 slits simultaneously, MUSE will, for the first time, "freeze" (at a cadence as short as 10 s) with a spectroscopic raster the evolution of the dynamic coronal plasma over a wide range of scales: from the spatial scales on which energy is released (≤0.″5) to the large-scale (~170″ × 170″) atmospheric response. We use numerical modeling to showcase how MUSE will constrain the properties of the solar atmosphere on spatiotemporal scales (≤0.″5, ≤20 s) and the large field of view on which state-of-the-art models of the physical processes that drive coronal heating, flares, and coronal mass ejections (CMEs) make distinguishing and testable predictions. We describe the synergy between MUSE, the single-slit, high-resolution Solar-C EUVST spectrograph, and ground-based observatories (DKIST and others), and the critical role MUSE plays because of the multiscale nature of the physical processes involved. In this first paper, we focus on coronal heating mechanisms. An accompanying paper focuses on flares and CMEs.