I've played for several hours now and already cleared 3-4 monster nests but I could not figure out how to destroy those. I got the bombs for so how to destroy the nests with it?
You can destroy Monster Nests just like in Witcher 2. Crafting or Buying Bombs and using them on the Nests.
In the Witcher 3 it works the same way you have to blow up the nests but with some new twists. Before you would go to the nest and press a button to drop the bomb.
Now in the Witcher 3 you can throw a bomb towards the nest or use a destructive sign at it to destroy it.
"look around the chamber until you spot the two smoking mounds lying either side of the pit. As instructed, use either bombs or one of your destructive spells to obliterate them" - Quote from Guide
The specific bomb to destroy nests in the Witcher 2 was the Grapeshot Bomb.
Ghouls Nests and Nekker Nests can only be destroyed with bombs. Get a Samum Bomb or Grapeshot Bomb (Most Common) and go to the Nest and will ask you to press a button to drop de bomb just like in Witcher 2. Some nests can be destroyed by using the bomb and Igni and some only with Igni.
Endrega nests (Eggs) can be destroyed using Igni and so can rats nests.
If you need to replenish your bombs just meditate with Strong Alcohol like Alcohest and they will refill along with your potions.
Grapeshot & Samum bombs can both be used to destroy nests...
You start with both of these recipes and can craft them yourself (you do not need a craftsman) once you find the materials. You can loot/harvest the materials in the starting area but saltpetre is not common however if you can buy it from a herbalist. Once you have crafted the bombs you can use them to destroy nests simply by pressing the action button/key when near a nest and Geralt will light and drop one in resulting in the nest exploding and you being rewarded with XP for destroying it a few seconds later.
Be sure to stick around for after the explosion as the monster nest usually drops valuable loot and crafting components.