To move your Simple DNS Plus installation including settings and data, you first need to backup your data and settings - click here for details.
Then install Simple DNS Plus on the new computer and restore the data and settings (see link above).
It is not allowed to use the same license key on more computers than licensed for - for any period of time, no matter how short this may be.
Therefore, if your Simple DNS Plus license is activated on the old computer, you need to remove Simple DNS Plus from the old computer (un-install or delete files), before you activate the license on the new computer.
If you need to run both the old and the new Simple DNS Plus instances during a transition period (for example if the new computer also has a new public IP address), you can use a trial license on the new computer during this time.