The linear momentum in inelastic collision is conserved the reason is following.

According to the Newton's second law of motion the rate of change of momentum is equal to applied force on the system. Now if we consider the two colliding bodies  to be a system then there is no external force and therefore the momentum needs to be conserved because the rate of change of momentum is equal to zero.

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Momentum is conserved, because the total momentum of both objects before and after the collision is the same. However, kinetic energy is not conserved. Some of the kinetic energy is converted into sound, heat, and deformation of the objects. A high speed car collision is an inelastic collision