Getting verified on Mastodon is relatively simple and straight forward. There's no official website or institution that you can go to and that will check if the Mastodon account you operate is really about you or not. Here it lies all in your hands. Actually it's a simple and clever way for a otherwise rather tricky problem. All you need is a website that is under your control. From there it's a rather simple three step process.
First login to your Mastodon account, go to "Edit Profile" and look for the "Verification" tab. Copy the code that is provided there for you. The code looks something like this:
<a rel="me" href="https://mastodon.example/@username">Mastodon</a>Usually that is not exactly the code that you want on your website because that way a visible link with the word "Mastodon" will be created. My recommendation is to use a code like this instead:
<link rel="me" href="https://mastodon.example/@username">Step 2. Now simply add your personal website address to your Mastodon profile under "Profile metadata". A secure address with "https://" is required here.
Proceed to Step 3. Go to the website you own and insert the above code somewhere within the <head>...</head> tag. But honestly where exactly you put that line of code doesn't matter so much. You can also put the line of code in the <footer>...</footer> tag. Of course some basic understanding of HTML is helpful but it's not really required. A website which describes the whole process of verification in even more detail can be found here. It was very helpful to me back in the day.
One last word that is often overlooked because it's almost taken for granted these days. The Mastodon verification only works with websites that already use "https://example.com" instead of the older "http://example.com". So watch out for the additional "s" in the address bar of your website. Your website needs to support the more modern, encrypted way of communication in order to work for this process.