You would simply download and install the program (using the internet) on your new computer when you get it. You can unlock it on the new computer and it will authorize. If you don't need to copy your data files to the new computer, you're good to go.
If you do want to copy your data files, run Logbook on the old computer and select Help > Open Data Folder. Then close Logbook (the data folder window will remain open). Copy or email the EclipseLogbook.SDF file from the data folder to your new computer and place it in the data folder there (use the same process as the old computer to open up your new data folder).