You may use your License for Media Center on additional personally-owned computers. If you have the Master License, it is good for Windows, Mac, and Linux. You definitely do not need to purchase an additional copy.
The simplest way to activate the license on additional computers is to install it and then enter the Registration Key when prompted. This will "restore" a copy of your license onto that computer.
However, if it has been less than 14 days since you activated your license on your original computer, though, you can instead simply copy the MJR file you were emailed over to that machine, and then double-click it to open it in MC. That will also license the additional computer (without "using up" a restore). Those MJR files expire after 14 days, though, so if it has been a couple weeks since you activated the first machine, you'll want to do it the "regular way" instead.
More detail:
https://wiki.jriver.com/index.php/Restoring_a_License