To move all your abbreviations from one Mac to another one you need to transfer your sets folder to the new computer.
Check the path of the "Sets folder" in Typinator's settings. Copy this folder to the same location on the other computer, then choose this folder as the "Sets folder" in Typinator's settings. You can also store the Sets folder on Dropbox to automatically synchronize it between multiple Macs. For more information, see the article "Sync Typinator sets with Dropbox
Hint: In the default configuration, Typinator stores the Sets folder in the Library folder, which may be invisible on your computer. To quickly open the Library folder, open the Finder's "Go" menu, then press the option/alt key (⌥), and the item "Library" appears in the menu.