You have a computer with a full retail copy (not pre-installed and not upgraded) version of Windows 7 or 8.1.
You upgrade it to Windows 10.
So, if you want to take that to a new computer, you would:
Remove the OS from the original computer. This would mean removing Windows 10, but you can't put Windows 7 or 8.1 back on it, because that's the license you want to transfer.
Install Windows 7/8.1 (the full qualifying version) onto your brand new computer. Activate it. You might need to call the hotline to activate, and explain to the agent that you are installing it on a new computer now and that the old computer no longer has it installed.
Upgrade it to Windows 10. Note that upgrades from previously-qualifying versions are only free until July 29, 2016. After that date, you'll need to buy an upgrade.