The one-dimensional transverse-field Ising and XY model are integrable. Their Hamiltonians can be diagonalized by a transformation to a system of free fermions. Their relaxation process after a quantum quench can thus be described exactly applying Wick's theorem and expressing the observables as Pfaffians. Although the results of this approach are exact, they do not give an illustrative description of the processes within the system after the quantum quench as well as of the time-dependent shape of the observables. This description can be given using the semiclassical theory for the relaxation process in the one-dimensional transverse-field Ising and XY model. The quasiparticles of the semiclassical theory are broken bonds, so-called kinks, which are created by the quantum quench. Their occupation number depends on the quench protocol and is conserved under the unitary time evolution. The conserved quantities of the system can thus be expressed in terms of the occupation numbers of the quasiparticles. Due to the momentum conservation the quasiparticles are created pairwise with opposite velocity for each wave number by the quantum quench at each site of the chain. The quasiparticles of a pair are entangled. After their creation they move ballistically through the chain with a constant velocity depending on the wave number and the parameters of the Hamiltonian after the quench. They do not interact with each other and cause the relaxation process by flipping a spin when the pass its position. For open chains the quasiparticles are reflected at the boundaries. Taking into account the position of each quasiparticle created by the quantum quench the values of the observables can be computed at each time with a high accuracy. For finite systems there is a quasiperiodicity of the observables caused by quasiparticles passing the same position several times in the course of time. The semiclassical theory gives an illustrative but highly accurate description of the relaxation process in the one-dimensional transverse-field Ising and XY model and is also capable of describing finite-size effects and the quasiperiodicity of the observables in finite systems. Besides it allows to construct the generalized Gibbs ensemble describing the stationary state of the system expressing the conserved quantities in terms of the occupation numbers of the quasiparticles.