How have people been able to isolate and identify nutrients in foods and their functions?!

The other two answers explained most of it already. As for their functions, once a food is found to contain certain nutrients and a group that consumes said food has good health or low rates of a certain type of cancer, researchers isolate what they think is responsible for this and begin testing, usually on mice or rats. They will induce a disease/condition, like giving the rats cancer or exposing them to high doses of carcinogens, then introduce the nutrient to see if that helps prevent/halt cancer production/formation.

Also, strawberries, some dried herbs and bell pepper all contain much more vitamin C than citrus fruit.

ph balance for vitamins,
chemical testing for amino acids and protein content etc..
its basic chemistry

Scientific experiments. Medical tests. Observation.