During the Vietnam War, the US government wanted to win the hearts and minds of the Vietnamese people. It means they tried to make the Vietnamese people like and trust them.
Imagine you want to make friends with a new person. If you are kind and helpful to that person, they will like you more. This is what the US government tried to do with the Vietnamese people. They did things like building schools, giving medical care, and supplying food to villages.
But at the same time, the US was also fighting against the Vietnamese people who were fighting against their government. This confused the Vietnamese people. They didn't understand why the US was helping them in some ways and fighting against them in others.
Winning hearts and minds is like trying to make someone chose you to be on their team. You want them to want you. The US government tried to make the Vietnamese people want them on their team. But it was hard for the US to win the hearts and minds of the Vietnamese people when they were fighting against them at the same time.