How to login and check my inbox mail of private relay apple id

You can’t and you don’t need to. Any email sent to your private address will be automatically forwarded to your regular email address.

This is a public forum; you should not post any personal information such as email addresses for the whole world to see. I’ve asked the hosts to mask your email addresses.

Scroll to the top of this page. In the upper right corner it says “get support” Click on Get Support.

They probably didn’t actually send it, then. You can check with Apple support by clicking on Get Support at the top of this page.

Check your spam folder.

Here’s more information→Hide My Email for Sign in with Apple - Apple Support

It’s called “junk” in some mail clients, and there are 2 messages in it.