Recently, the discovery of room-temperature superconductivity (SC) was experimentally realized in the fcc phase of LaH$_{10}$ under megabar pressures. This SC of compressed LaH$_{10}$ has been explained in terms of strong electron-phonon coupling (EPC), but the mechanism of how the large EPC constant and high superconducting transition temperature $T_{rm c}$ are attained has not yet been clearly identified. Based on the density-functional theory and the Migdal-Eliashberg formalism, we reveal the presence of two nodeless, anisotropic superconducting gaps on the Fermi surface (FS). Here, the small gap is mostly associated with the hybridized states of H $s$ and La $f$ orbitals on the three outer FS sheets, while the large gap arises mainly from the hybridized state of neighboring H $s$ or $p$ orbitals on the one inner FS sheet. Further, we find that the EPC constant of compressed YH$_{10}$ with the same sodalite-like clathrate structure is enhanced due to the two additional FS sheets, leading to a higher $T_{rm c}$ than LaH$_{10}$. It is thus demonstrated that the multiband pairing of hybridized electronic states is responsible for the large EPC constant and room-temperature SC in compressed hydrides LaH$_{10}$ and YH$_{10}$.
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