Like all files, they need a specific type of program to open them.
.dbx is a database of your emails and used by Outlook Express (your previous email client).
You need to import them into your present email client. That may or may not be easy depending on which email client you are using on your new computer.
Please post details of new computer and client in use and then someone who knows how to import them to your new email client will be able to help you.