A field scout checks fields, monitoring plant health and measuring levels of insect infestation, and reporting their findings to a farmer or agronomist. Basically, what you do is drive around out in the country, enter fields to make and record observations, then report your findings to decision makers.
I scouted cotton and alfalfa, so the tools I used most frequently were a canvas bug net (see picture) and a 16x loupe. The net handle was graduated (by hand) to easily measure cotton growth. I would also use a folding canvas tarp, but only for very tall cotton. Other items used by field scouts include a soil probe (see picture) for taking soil samples, a yardstick for measuring plant growth (not needed if you graduate your net), some rugged boots, and a broad-brimmed hat.
As you might expect, a bug net is used to sweep for bugs. Exactly how this is done differs from crop to crop, but the basic principle is that you sweep the plant and record the number and type of insects found.
A loupe is necessary because some insect pests are too small to see otherwise. For example, it's not very easy to see spider mites without magnification.