The video is hilariously wrong.
However, the principle of laser ranging is more or less right, and it does require the reflectors left behind by the astronauts on the Moon. It's just that the physics and technology involved are far beyond just pointing a toy laser at the Moon.
Project APOLLO (Apache Point Observatory Lunar Laser-ranging Operation) is actually doing this.
You need a fairly large telescope to start with - both for collimating the outgoing light pulse, and for receiving as much of the reflection as possible. APOLLO uses a 3.5 meter telescope at the Apache Point observatory.
You need a laser that can generate a high-energy light pulse that is very short. The pulse is injected into the telescope's optical train and sent to the Moon. This is not a laser pointer; it's a high power Q-switched laser for research, a device the size of a refrigerator.
On the receiving end, you need a very good detector also plugged into the telescope. Of the many, many photons sent to the Moon in the pulse, only between 1 and 5 photons make it back down to the detector. You need a detector that can tell the extra 1 to 5 photons from the background noise of light coming from the Moon.
Using this technique, the distance from Earth to Moon can be measured with very high precision.
This is the APOLLO system in action:
Regarding observing human artifacts on the Moon with terrestrial telescopes, I wish it was doable, but it's not. Again the YouTube video is wrong.
The biggest telescope accessible to amateurs has an aperture (diameter) in the range of 1 meter or a little bit larger (the aperture of the largest amateur-owned telescopes currently). The resolving power of a telescope (the size of finest details discernible) depends on aperture - if aperture is measured in mm and the resolving power in measured in arcseconds, then the formula is:
resolving power = 100 / aperture
So a 1 meter telescope has a resolving power of 0.1 arcseconds.
The distance from Earth to Moon is 384000 km (3.84 * 10^8 meters). With a resolving power of 0.1 arcseconds, the size of the smallest detail discernible on the Moon is:
detail size = distance_from_Earth_to_Moon * arctan(resolving power)
detail size = 3.84 * 10^8 * arctan(0.1 arcsec) = 186 meters
Anything smaller than 186 meters would be blurred into a single dot. There's nothing we've done on the Moon that's as big as that. It's not possible to see traces of human activity with amateur-level telescopes, even with extremely large meter-class dobsonian telescopes. Even with professional telescopes, we just don't have the aperture yet to resolve that kind of detail.
However, satellites in orbit around the Moon, such as the LRO, were able to image traces of the Apollo missions. That's because they are a lot closer to the site.