In Silent Spring, Carson is at pains to show that humans are intimately connected to nature and are not floating above it. What happens to the natural world has a profound effect on us.
Carson argues that we need to control pesticide use, because the damaging effect it is having on the environment will seep up to us. She asks us to envision a "silent spring" in which the world is utterly quiet because all the songbirds have been killed by pesticides. If the songbirds can be destroyed by pesticides, so can we. We are all part of one ecosystem.
Carson uses what was cutting-edge medical research at the time to show that pesticides cause health problems, such as cancer, in humans. If we humans are to survive and thrive, we need a healthy ecosystem.
The dropping of the atom bomb on Hiroshima and Nagasaki in 1945 continued to make a deep impression on the American public even in the early 1960s, when this book was published, because of the immensity of the destruction. Humans had to come to grips for the first time with the fact that we had created a technology powerful enough to end life as we know it. Carson makes repeated references to the possibility of atomic holocaust and the damage caused to living creatures by its invisible radiation as she builds her case that modern technologies, even those which produce something seemingly as invisible as pesticides in soil and water, have immensely destructive potential.
Yes, I agree that humanity is deeply enmeshed in and dependent on the natural world and that we must treat it with care if humans are to survive.