I've never figured out how some people do that . . .
It used to be pretty easy: Open a YouTube account, open an Adsense account under the same Google account, link the two, and decide which videos to monetize.
Nowadays it's basically the same process except that the YouTube account has to be "verified" first, and your videos have to achieve a certain number of hours of viewing. The specific number of hours has changed a few times. YouTube also has been known to approve accounts that are rapidly accumulating the view time even if they haven't yet gotten there. They basically want popular accounts.
YouTube's "Verification" process usually means letting them send you a text message. It's just a way for Google to get your cell phone number, as far as I can tell.
If the Google account has already been verified by some other means, however (such as by postcard for a Google My Business account or if you already have an Adwords, Adsense, or Google Search Console / Webmaster Tools account), they'll sometimes offer the option of verifying over a landline with a recorded voice call -- provided it hasn't been used to verify too many other Google accounts.
One way or the other, the objective of verification seems to be for Google to get another piece of information about you.
Once the YouTube account is "verified," the viewing hours start adding up. When they get to a certain point, Google will either invite you to monetize it, in which case approval is basically automatic; or you can apply yourself to monetize it, in which case they'll check to make sure your videos have the requisite number of view hours and that they don't offend whatever sensibilities Google happens to be observing that particular day.
If all your videos are about things like flying, car repairs, home improvement, or other "safe" topics, your approval should be quick as long as you have the requisite hours. If they're about hunting, guns, religion, or politics, then not so much. They don't want controversial videos monetized.
Once you're approved for monetization, you open an Adsense account if you don't already have one. They'll need the standard tax information that any 1099 payer needs. Approval will be automatic if they've already approved your YouTube channel for monetization, but it may still take a few days.
Once the videos are monetized, money is earned when people click the ads. They may also have pay-for-performance ads now, which would require that viewers do more than click, but would generate more money. If you're hugely popular, they may also have pay-for-impression ads available. They pay based on the number of times an ad appears, regardless of whether people click it. I really don't know what's available now other than pay-per-click, which were the most common ads that Google offered to Adsense publishers the last time I checked. I haven't set anyone up with monetized YouTube for a while, however, so things may have changed.
There are better ways to monetize videos without Google and YouTube, but they're a lot more work and require that you have your own hosting, some basic competency in HTML5 and JavaScript, decent editing software, and scads of storage and bandwidth or a business account with Vimeo or some other video hosting platform that allows embedding. They also require that your site achieve some popularity of its own. YouTube's biggest advantage is its popularity and ubiquity. That's where most people go for HOWTO videos. But if you've established enough of a name for yourself that people will specifically seek out your advice, building your own site and monetizing it with a commission-based platform like Amazon Associates will probably make more money for you than Google's PPC model.
In a nutshell, Google is fast and easy if your videos are popular and utterly inoffensive. It also will generate passive income for you. But there are more lucrative ways to go about it -- especially if your videos are about merchandise that you can hawk -- once you've established a name for yourself as a go-to source for information about your particular area of expertise.
Rich