It isn't straw we chew, it's grass.
Straw is made from the stems of wheat, oats, or barley that's been cut and baled to provide bedding for animals after the grains have been harvested.
What you see farmers chewing is grass--tall, tasseled prairie grass--which is liable to be growing in yards and pastures out in the country. There isn't a specific reason why we chew it. We just do. It's kind of like chewing gum. Many tall prairie grasses have a sweet, refreshing taste--one of the kind I like to chew, called big blue stem, actually tastes a bit like watermelon. So if you're working outside and it's hot out and you don't have any gum, a long piece of grass not only gives you something sweet to chew on but it can also (ever so slightly) re-hydrate you. In the very least, it breaks the monotony of farm work.