The concept is simple. The verification system will provide you with an HTML file or Meta Tag that you insert into your code and upload to your site.
Then you validate it by running their crawler to verify that you own that website. The image below is the Meta Tag version.
I also use the HTML file too.
There are several other ways to verify your site but HTML and Meta Tags are the most convenient. Yahoo Search merged with Bing some years ago so the Meta Tag y_key is no longer in use.
Google Search Console
"Verification is the process of proving that you own the site that you claim to own.
We need to confirm ownership because once you are verified for a site or app you have access to its Google Search data, and can affect its presence on Google Search." - Google
Bing Webmaster Tools
"Webmaster Help & How To provides you with assistance for the Bing Webmaster Tools as well as with guidelines on how to be successful in Bing search." - Bing
Pinterest Site Verification
"You can claim websites you own. To claim your website, you'll need to add a meta tag or upload an HTML file to your website's HTML code. - Pinterest