What sort of methods/theories did the author employ, and what sort of sources did the author rely on?

Rachel Carson was a biologist who worked for 16 years (1936-52) for the Fish and Wildlife service, so she had a strong—and long—background in the science she documented for her book. She had been worried about the use of pesticides since the 1940s. Because of her concerns, she had made a point of following court cases that fought the use of pesticides. In one case, for example, people in Long Island sued to prevent their land from being sprayed with DDT to control the gypsy moth. 
One theory she relied on was biomagnification, which argued that the cumulative effects of toxins as they build up in the environment and the human body are worse than the sum of their parts. She also relied heavily on the medical field, tying together medical and biological research to argue that pesticides caused cancer.
In the four years before publishing her book, Carson did a vast amount of research. She did much of it at the NIH (National Institutes of Health) and the Library of Medicine. She examined thousands of scientific papers on pesticide use and related issues. She interviewed scientists. She also discovered that medical researchers were studying the link between pesticides and cancers and incorporated their work into her research. She interviewed doctors. She used the work of Wilhelm Hueper, who had determined that many pesticides caused cancer. She also researched history to document her link between toxins and cancer, even using research on cancer in chimney sweeps from the eighteenth century, which connected their scrotal cancer to constant contact with soot. 
Carson's method was not simply to rely on scientific reports and papers that would be difficult for the average person to understand. She used newspaper reports, personal interviews, and court cases to show how pesticides were impacting everyday life. Trying to build sympathy for her case, she showed the devastating effects of pesticides on the beloved robin, and on a living symbol of the United States, the eagle. She also showed how pesticide use had harmful effects on ordinary humans. She likened pesticides to radiation poisoning from atomic bombs: invisible but devastating. That would have been a potent parallel for people still coming to grips with atomic power. She very pointedly aimed her research at persuading people that their own lives were being damaged—or could be damaged very easily—by the overuse of pesticides that entered the food chain and ended up in the human body.