President Nixon tried to give “ownership” of the war back to the government of South Vietnam because he saw it as a way to extricate the US from the war, getting “peace with honor” for the US.
By the time that he became president, Nixon had decided that the US needed to get out of the war. However, he did not want to be seen as giving up or surrendering. Therefore, he had to find a way to leave Vietnam without appearing to surrender. Vietnamization seemed like a good way to do this. It made sense to require the Vietnamese to fight their own war. It also let the US leave without actually surrendering to the North Vietnamese.
In this way, Vietnamization was a good cover that would allow the US to withdraw without appearing to give up the fight.