Every once in a long while you may need to open another user’s mailbox to look at a suspicious email or delete something someone sent to someone else that they were not supposed to receive.  Both are legit and do happen.  Here’s what to do:

To give the user “Administrator” and “Administrators” group access to all exchange mailboxes run this command from Powershell with Exchange Add ons loaded:

Get-Mailbox -ResultSize unlimited -Filter {(RecipientTypeDetails -eq 'UserMailbox') -and (Alias -ne 'Administrator')} | Add-MailboxPermission -User administrator -AccessRights fullaccess -InheritanceType all

Now log into Exchange Webmail with the Administrator account and follow these steps to open the user’s mailbox:

1. Click your name in the upper-right corner of the window to open the Open Other Mailbox dialog box.
2. Under Select Mailbox, enter the name or alias of the mailbox you want to open and click Open.
3. If more than one match is found, a list of possible matches will be displayed.
4. Click the name of the mailbox you want to open.

References for detailed reading:
Give an Admin access to Exchange mailboxes
Open another Exchange mailbox