As a beginner, how does IP lookup actually work?

Hi everyone, I'm quite new to networking concepts and I'm trying to wrap my head around how IP addresses work, especially after seeing the discussion about 'my public IP is confused?'.

My main question is about how services like 'whatismyip.com' or 'ipinfo.io' actually perform an IP lookup. What's happening behind the scenes when I visit one of these sites? Is it DNS? Is it something with my router? I'm trying to understand the basic mechanics.

Specifically, how do they determine my public IP address, and what information are they using to do that? Any simple explanation would be super helpful for a total noob like me! Thanks in advance!