So this has nothing to do with hay or actual straw, but it is an unofficial vote at the early stages of a campaign to indicate which candidates are preferred.
So if we conducted a straw poll here with the Agree Team before deciding which ice cream flavors to buy for our imaginary ice cream party, we would ask: do people like chocolate, vanilla or strawberry better? This would give us a better sense of what side people are on and what we would plan on buying.
More importantly, in politics, a straw poll can help determine which presidential candidate is likely to win. It’s not the official vote, so there is time for public opinion to change (just like changing the choice of our favorite ice creams).