We give an exact form of the density matrix of the spin-1/2 Kitaev spin liquid represented in terms of spin operators and study the entanglement structures of the Kitaev honeycomb model within the spin framework. We show that the density matrix is naturally organized by equivalence classes of string operators associated with the underlying gauge structure of the model. With the explicit form of the density matrix, plus the exact Gauss law of the emergent gauge theory and the exact 1-form Wilson symmetry in the Kitaev model, we demonstrate the existence of the underlying symmetry-resolved block-diagonal structure of the reduced density matrix, which gives rise to the extensive degeneracy in the entanglement spectrum. The block-diagonal structure is then proven to be responsible for the separability of the entanglement entropy into the gauge and matter parts. Furthermore, we extend the formalism to subsystems with an odd number of lattice sites, revealing a relation between the entanglement spectrum and the fermion parity that is seldom mentioned in the literature.