It's not possible to mount the databases from one instance of Exchange onto another one just like that. Actually to clarify, it is possible within the same Exchange org and might be required in some DR or reinstall scenarios, though I'd suggest it's not good practice as a general use thing.
As for migrating mailboxes, you can export them to PSTs then import them again. I'm not convinced this scales well for lots of users but it's there.
Grant yourself rights to do it (I'm assuming you're logged in as administrator) and then restart the exchange shell.
New-ManagementRoleAssignment –Role "Mailbox Import Export" –User Administrator
Ensure the target that you're exporting the mailboxes to has write permissions for the Exchange Trusted Subsystem group (important if you're doing this over a network to a share).
To export all mailboxes from an Exchange Organisation:
foreach ($i in (Get-Mailbox)) { New-MailboxExportRequest -Mailbox $i -FilePath "e:\$($i.Alias).pst" }
These commands are a bit "approximate" as I don't have an Exchange 2010 server handy to double-check them with but they'll certainly get you into the ballpark.
As for what to do on Exchange 2013 once you've got the mailboxes, there's the New-MailboxImportRequest
cmdlet. Not played with this on Exchange 2013 so I'll just point you at the documentation for it. Needless to say you're basically importing the results of the earlier script into new mailboxes at this point..