Each email from your group has some links.
Here is an example of an individual email, with the links at the bottom (the links will be formatted differently if the message from your group is plain text, rather than formatted):
View/Reply Online brings you to that very message. Group Home brings you to the group's Home page. You can use either link to find your group's pages, and from there the Messages list.
Note: if the Privacy section of Settings page of your group has Visibility set to one of the private (not public) archive choices, then members will have to log in to Groups.io in order to access the Messages. In the upper right of a Groups.io page they should see the Log In link:
As a member of the group you already have an account, so you don't need to use the Sign Up form. You may not have a password for your Groups.io account yet, and if that's the case the Log In form will give you a couple of alternatives:
Either highlighted link will send you an email message that will let you log in to the site. The don't have one yet? message will have a link that will bring you to your account page, where you can set/change your password. The Email me a link to log in message will have a link that brings you back to the page you started at.
Once logged in to the site you will be able to see the group's Messages page:
Note: You may not see some items, such as Subgroups or Directory, depending on choices the Owner has made in Settings. Here's what you would see in Groups.io Beta group:
You can view messages by Topics (threads), Messages (individual within threads), Expanded (all messages chronologically without threading). For the messages options, Groups.io remembers where you left it last time (topics, etc, but not polls). This is also where you can see existing polls.