Entrenched = hard to remove or get rid of, such as a person who cant be fired from a job, or an idea that people refuse to stop believing. ("The idea that money is the most important thing is entrenched in the minds of many people, even though they would be happier if they made less money but worked less "). It is from the trenches that soldiers dig to make it hard for an enemy to force them to retreat.
Nestled in is from the root nest, like a bird's nest. It is to make yourself warm and cozy, like under the covers in bed,