A linear Boltzmann transport (LBT) Monte Carlo model has been developed to describe jet propagation and interaction with the quark-gluon plasma (QGP) in relativistic heavy-ion collisions. A complete set of elastic-scattering processes and medium-induced gluon emissions based on the higher-twist formalism are incorporated for both jet shower and medium recoil partons. It has been employed to describe experimental data on large transverse momentum hadron and jet spectra, correlation and jet substructures in high-energy heavy-ion collisions. We document in detail the structure of the model and validation of the Monte Carlo implementations of the physics processes in LBT, in particular, the inelastic process of medium-induced gluon radiation. We carry out a comprehensive examination of the jet-medium interaction as implemented in LBT through energy loss and momentum broadening of a single hard parton, the energy and transverse momentum transfer from leading partons to medium-induced gluons and jet-induced medium excitation, and medium modification of reconstructed jets in a static and uniform medium. With realistic and event-by-event hydrodynamic medium in heavy-ion collisions, we compute and compare with experimental data on the jet cone-size dependence of the single inclusive jet suppression at both the BNL Relativistic Heavy-Ion Collider (RHIC) and the CERN Large Hadron Collider (LHC), the dijet asymmetry at the LHC and γ-jet correlation at RHIC. Effects of medium-induced gluon emissions and jet-induced medium excitation on jet observables are systematically examined. Rescatterings of the radiated gluons and recoil partons with the QGP are found essential to account for the enhancement of soft particle yield toward the edge of the jet cone.