How can immigration help the economy?

I don't have the formal training in economics to give you the best answer possible, but I can give you my understanding of the situation.

First there are a few facts to consider.

(1) When unemployment is under 5%, as it is now, it is considered full employment.

(2) Without immigration, the American population is relatively stagnant.

(3) The number of people in the workforce is declining, largely due to baby boomers retiring.

It is obvious that every time the population goes up by one, there is one more person buying goods and services in the county. A larger population buys more food, housing, transportation, televisions, food, entertainment, medical care, legal services, banking services, etc. all of which create more jobs and taxes, so the economy grows.

It is also my opinion that in general people moving to a new country to create a better life are more likely to work harder to create that better life. The videos in the link you posted give some examples of that happening, but I would have to do some research to find if there are any formal studies or statistics to support my opinion. I know there will be some exceptions but I do not believe they are widespread.

I'll start to consider this when the EU and other First World countries start doing it. The function of a government is NOT to care about people in other countries, it's to protect and help its own citizens.