Why did secureserver net godaddy access my gmail account?

According to my web searching, secureserver.net is a domain that's owned and operated by GoDaddy, and secureserver net GoDaddy provides email services to its customers. For example, GoDaddy customers can set their SMTP server for outgoing email to smtp.secureserver.net, and use that to send outgoing email. Similarly, GoDaddy customers can read their email there (via POP or IMAP). It appears that GoDaddy also occasionally sends emails to its customers from the secureserver.net domain. GoDaddy may also use it for other purposes as well.

Did you perhaps enter your gmail account and gmail password into the secureserver.net website, or provide it to GoDaddy? Perhaps their software on that domain will check your Gmail for you if you provide it with your Gmail username and password (I am speculating now, as I am not a GoDaddy customer).

If you want to know for sure why secureserver.net accessed your Gmail account, I suggest contacting GoDaddy support.

P.S. As @tylerl suggests: change your Gmail password now.

Change your security question, too. Then, go to the Activity Monitor (click on "Details" next to the thing saying "Last account activity: X minutes ago at this IP (xx.xx.xxx.xx)") and click "Sign out all other sessions".

Also, check the list of secondary email addresses and the forwarding options for your Gmail account to make sure that your Gmail is not being forwarded somewhere (a favourite ploy of hackers, if they gain access to your Gmail account, is to set up a secondary address or forward your email to themselves, so they can regain access later using a password reset).

While you're at it, consider enabling 2-step verification); once that is activated, logging into Gmail will require both your Gmail password and access to your phone. This will help prevent cases where others try to log onto your Gmail, since they won't know the verification code that is sent to your phone.

Also, you could check the Gmail Activity Log (Activity Monitor), to see if it has any further details about what accesses have been made to your Gmail account.