The video opens up with this very incorrect statement:
"In the United States, the Supreme Court is the highest court, given the final say on what laws really mean, and if they're cool with the Constitution. Well, this power was not given, but taken. Back in the day, the Supreme Court ruled it is the duty of the court to say what the Law is. And everyone went with it."
This is what the video starts out with, and it is a talking point people sometimes bring up. It's also inaccurate. Note how he claims this power "was not given, but taken" and that it only happened when the Supreme Court ruled it had that power.
The power of judicial review, the declaration if something is constitutional, is straight there in the Constitution:
"
The judicial Power shall extend to all Cases, in Law and Equity, arising under this Constitution, the Laws of the United States, and Treaties made, or which shall be made, under their Authority..."
Obviously, the question of whether a law or something else is in agreement with the Constitution is a case in law. So it's there in the Constitution. When the Supreme Court asserted it had that power in Marbury v. Madison, it was just a recognition of the fact, not a taking of it.
This interpretation was agreed upon even before there was any Supreme Court. Various people at the Constitutional Convention said courts would have the power of judicial review, and afterwards (during the ratification process) people publicly said it at things like ratifying conventions or in print.
The Federalist Papers, which were written and distributed to try to convince people, particularly in New York, the Constitution was a good idea so they'd ratify it, explicitly declared "whenever a particular statute contravenes the Constitution, it will be the duty of the judicial tribunals to adhere to the latter and disregard the former."
For that matter,
opponents of the Constitution said judges would have that power, perhaps most prominently Brutus, a pseudonymous writer who wrote a set of essays attacking the Constitution to try to convince people not to ratify it. He didn't say the Constitution didn't grant that power.
He asserted it did, and argued it was a bad idea because it would make judges more powerful than the legislature ("If, therefore, the legislature pass any laws, inconsistent with the sense the judges put upon the constitution, they will declare it void; and therefore in this respect their power is superior to that of the legislature.") Patrick Henry, who also opposed the Constitution--a fact a lot of people don't know about--affirmed this in a ratification debate, though his problem was ironically the opposite--he worried that if federal judges were the ones who made the decision, they'd just rubber stamp everything even if it was unconstitutional (
see here: "Yes, sir, our judges opposed the acts of the legislature. We have this landmark to guide us. They had fortitude to declare that they were the judiciary, and would oppose unconstitutional acts. Are you sure that your federal judiciary will act thus? Is that judiciary as well constructed, and as independent of the other branches, as our state judiciary? Where are your landmarks in this government? I will be bold to say you cannot find any in it. I take it as the highest encomium on this country, that the acts of the legislature, if unconstitutional, are liable to be opposed by the judiciary.")
Both supporters and critics of the Constitution agreed that the federal courts, and especially the Supreme Court, would have the power to declare laws unconstitutional. They disagreed on whether it was a good way to set up the government that way, but they agreed it's how the government would be setup.
So the video's claim is simply inaccurate. Maybe I'm focusing too much on something that only lasts a few seconds... but it's literally what the video starts with.