The object is pushing into the earth due to its non-zero velocity, exerting more than the usual force of gravity on the earth, so the reaction force on the object is larger than just the force of gravity for non-zero velocity. This decelerates the object until its velocity is zero, and the reaction force on the object is just the normal force counteracting gravity.
It doesn't happen spontaneously on contact, but it usually happens rather fast - unless the earth is so soft at the impact spot that the object can bury into the ground.
The source of the force the earth exerts on the moving object is due to the earth resisting the object moving into it. Two solid objects cannot occupy "the same space", and when one is moving such that it would end up inside the other, the other resists by exerting force. The origin of that force is Pauli exclusion and electromagnetic repulsion between the electrons of the atoms that constitute the two solids. For a more detailed discussion on why things can't move into each other, see Why doesn't matter pass through other matter if atoms are 99.999% empty space?