After the mission "Trouble With Harry" the first set of Symbiote Nests will start showing up on the west side of the city, they are massive symbiote towers that stretch over the building of the city and have a sinister red glow around them.

The second and final set of them will be unlocked after the mission "It's All Connected".
 
The way you take these down is by setting a timed sonic burst charge on the red "bulbs" that are at the base of these giant structures, after setting them you need to defend them for a few minute until they are ready and once the time ends the nest will be destroyed. Nests will have either 1 or 2 bulbs at their base and 1 is the much more favorable of the two. If there is 1 bulb you can set the charge and then stay near it, kicking any Symbiote that comes near off of it and last like that for the duration of the timer.If there are 2 bulbs the strategy will change slightly, you will need to defend both of them at once as they share a health bar, this means you will want to try to stay in the middle of them and react to incoming Symbiotes. If you see a large group of them on once side deal with them but if you get notified that the other bulb is being attacked (With a red !) zip over to the enemy doing it with  and focus on clearing the enemies on that side out.If a enemy is attacking a charge hitting them just once with webs or a attack will stop them from doing it, along with this while taking down these Nests you can make good use of the Sonic burst gadget. Any Symbiote it hits will be stunned temporarily and if you have it fully upgraded once it hits something it will shoot out 3 smaller proximity based Sonic Bursts that will trigger with anything walks over it.After the timer expires the nest will dissolve and it will be cleared, after doing this to all nests you will unlock the trophy.In general in each area, there are:

Harlem - 2 Upper East Side - 2 Upper West Side - 1

Financial District - 1 Chinatown - 1

Williamsburg - 1 Little Odessa - 1 Astoria - 1

Symbiote Nests  are the gathering places of Symbiotes that Venom created, and you must destroy them to prevent their development.

Cleaning Nests is done in such a way that you place a Sonic Burst in the heart of Nest and then you have to take care of it for 1:30 to 2:30 and cannot let the Symbiote harm it. When the charging time of Sonic Bursts is over, the heart will explode and the area will be cleared from Symbiotes.Access to Symbiote Nests is possible after Main Quest 26, but to access them all, you must complete Main Quest 28.

All Symbiote Nest Locations

Harlem

Symbiote Nest 1

Symbiote Nest 1

Symbiote Nest 2

Symbiote Nest 2

Upper East Side

Symbiote Nest 3

Symbiote Nest 3

Symbiote Nest 4

Symbiote Nest 4

Upper West Side

Symbiote Nest 5

Symbiote Nest 5

Financial District

Symbiote Nest 6

Symbiote Nest 6

Chinatown

Symbiote Nest 7

Symbiote Nest 7

Williamsburg

Symbiote Nest 8

Symbiote Nest 8

Little Odessa

Symbiote Nest 9

Symbiote Nest 9

Astoria

Symbiote Nest 10

Symbiote Nest 10