The following PowerShell snippet can be used to to export the results of a message trace, unfortunately at the time of writing the Office365 portal does not allow you to do this in the GUI.
Connecting to Office365 via Powershell
$UserCredential = Get-Credential
$Session = New-PSSession -ConfigurationName Microsoft.Exchange -ConnectionUri https://outlook.office365.com/powershell-liveid/ -Credential $UserCredential -Authentication Basic -AllowRedirection
Import-PSSession $Session
Exporting messages based on the sender address
Note the dates and in the American format mm/dd/yy
Get-MessageTrace -SenderAddress [email protected] -StartDate 03/31/2016 -EndDate 04/07/2016 | Export-Csv D:\messagetrace.csv
Exporting messages based on the recipient address
Note the dates and in the American format mm/dd/yy
Get-MessageTrace -RecipientAddress [email protected] -StartDate 03/31/2016 -EndDate 04/07/2016 | Export-Csv D:\messagetrace.csv