"Matching the Hatch" as a philosophy and methodology in the sport of fly fishing has been around for a very long time. The phrase became popular in North America when Ernest Schweibert published his book of the same name in 1955. It was his first of many books and was written when he was 24.
The match the hatch method of fishing is reliant on intimate knowledge of how trout function, and how the environment that they live in functions. It involves the study of how trout feed, and their relationship to the various prey organisms that exist in their environment. It also entails knowledge about how these same organisms live and behave to survive. This approach to fly fishing is actually a study of biology at many levels.
The theory concludes that trout tend to be very efficient feeders and often target organisms that tend to be pretty small in comparison to the trout themselves. There are calories gained in each acquisition, but not in large amounts. Each trout has to be careful not to expend more energy than is gained from every morsel, lest the fish starve to death.
In most stream environments, trout feed primarily on aquatic insects and crustaceans. They belong to the phylum Arthropoda, which is adapted to a wide variety of aquatic habitats. Crustaceans are critters whose skeletons are on the outside of their bodies. They are well designed for life on or in the bottom of rivers and lakes. They spend most of their lives crawling around there where they are often seen by game fish but rarely by humans.
Insects and crustaceans differ in one very important aspect. Crustaceans such as crayfish don't have a phase where they leave the water and become air breathing as many aquatic insects do. In fact, most aquatic insects not only have an aquatic and a terrestrial stage, they also have swimming and flying stages as well. This transition from an aquatic juvenile insect to an adult flying insect is called a "hatch," but is really a metamorphosis from one stage of life to another. However, this transformation from one stage to another is probably what started the "match the hatch" philosophy of fly fishing. Aquatic stages of insects in their true environment are hard to see and study. Even more difficult to observe is how trout react to them, or feed upon them. As soon as insects get above the surface of the water, they become more visible to humans. So do trout that are preying upon them.
Today a hatch can be anything that makes a specific kind of food available to fish in such a way that predator fish key on it. It may in fact be a hatch, such as insects emerging.