You find out the way to destroy the Nekker nests by obtaining enough knowledge about the Nekkers, either by reading a book about them (available in Flotsam) or by fighting them. If you're at knowledge level 3/3 (character screen -> Attributes -> Knowledge) there will be the following text about the Nekkers
Witchers can user their full range of
skills and techniques when fighting
nekkers. Signs, bombs, poisons and
oils that augment bleeding are all
extremely effective against the
beasts. Nekker nests are best
destroyed with Grapeshot bombs.
To build a Grapeshot bomb you first need the schematic, which is available from the elf Cedric. Just look out for the craftsman icon on the map in Lebinden to find him. After you bought the schematic you need to meditate and then build some Grapeshot bombs:
Thanks to Justin for the hint on how to destroy the nests.