After payment is complete, Paypal will give you a Paypal transaction ID (like 1D3342343468954574Z) or a receipt ID (looks like 1234-1234-1234-1234).
This ID will be your license activation key.
Please select and copy the transaction ID from Paypal page and paste it into plugin's activation code field in LR's plugin manager
to proceed with plugin activation. Please save your Paypal receipt, you might need it for activation in future, in case you reinstall
Note that the license is not dependant on your Lightroom serial number, you can even activate plugin on a trial version of Lightroom.