There are a few ways to approve messages, reject or delete messages needing moderation. 

As owner or moderator, you will receive an approval email for messages needing moderation.

- To approve this using email, reply to this message. You do not need to attach the original message, just reply and send.

- To reject this message using email, and to have the sender notified of this, forward this message to:

groupname+reject+number@groupname

- To delete this message using email, forward this message to:

groupname+delete+number@groupname.

NOTE: The pending message will expire after 14 days. If you do not take action within that time, the pending message will be automatically rejected.

This is the easiest way to moderate.
- In your email, just click to reply to the email (the notice, not the member) and it will be automatically approved. No message is required, just an empty reply email. 
You will be replying to the automatically address for that message ( groupname+approve+message number@groupname

(You must be using your email address associated with the account (i.e. your account ID). 

If you wish to edit the message, you need to do it online.

-  Online, go to https://groups.io/g/GroupName/pending (or Admin > Pending.) 
From there, there are two options. 

1) Select the posts using the check boxes to the left of them and click the Approve (or another) button.

or

2) Click on the first one to open it, then use the buttons at the bottom to Approve, Edit, Reject, Delete, etc. 

Once you take care of the first, the next one will automatically open.

Problems:

Email:  If another moderator or owner has dealt with the incoming message already, you will get a rather cryptic reply email. However, it will say:  500 This message has already been approved or rejected.

Online:  If you have followed the link but aren't seeing an opportunity to approve the messages, it's probably because you haven't logged in. Login instructions can be found at https://groups.io/static/help#password As long as you don't delete your browser cookies, the login will persist for 30 days.

Once you finish that, you should see a status badge at upper left (look for the Home button), confirming that you are logged in with a Moderator or Owner account. At that point you can go to Admin>Pending and perform your moderation duties online.

If another moderator or owner is dealing with the message, you will see "claimed" on the message. You can just back out.