If you are not backing up your old computer to transfer everything to your new one, but starting up fresh with just the most important folders/files transferred, you would have to transfer the FrostWire library manually. Gladly it's easy :)
On your old computer's FrostWire, go to Tools > Options > Library
Note all the folders your current FrostWire's Library is indexing.
TIP: Go to Tools > Options > BitTorrent and make sure your Default Download Folder is on the list. If not, add it too if you would like :)
Manually copy the folders to your new computer, wherever you want - they don't have to be in the same location.
Finally, after you've installed FrostWire on your new computer, go to Tools > Options > Library again and add all the folders back to your Library. Your new FrostWire will index them all again.