First, download the correct version of BeerSmith for your operating system.
http://beersmith.com/download-beersmith/
Once it's downloaded and installed, open BeerSmith and go to the Help tab.
Click on "Recover Key" and follow the instructions.
Once you've installed your key, you can then copy and paste the /Documents/BeerSmith2 directory from your old computer to the new one. This will install all of your recipes, preferences and profiles onto the new computer.
As with anything involving data, make backups before you proceed.