Mastodon's verification system is mainly for people and organisations who have their own website. It makes your Mastodon profile's link to your website turn green, proving you are the owner of the site.
Here's how to verify your account:
鉃★笍 https://fedi.tips/how-do-i-verify-my-account
If you've tried doing this and it didn't work, see the troubleshooting section of the guide ("My website address won鈥檛 turn green!") as there are lots of possible ways you might fix it.