Waht he is probably talking about is a feature that needs to be setup which is to leave a copy of messages on the server which will then make the mail available for your other machines.
To do this cna you open Thunderbird and select Tools from the top menu, then choose Account Settings from the drop-down list.
Next can you select Server Settings from the left window and in the Server Settings section of the right window can you place a check in the box labeled Leave messages on server then click OK and your mail will be available to your other machines.
Let us know if you have any issues
Let me know if you need any help on or after Sunday