Sense 1: "The other side" is... just the other side of the road. The joke is that you were given a joke set-up question (and, if encountering the joke for the first time in your life, perhaps put some thought into trying to puzzle out the correct joke-answer) but then the punch line was the straightest possible non-joke answer.

Sense 2: (the double-meaning) "The other side" is the afterlife. The chicken is courting death by crossing the road. I dunno, maybe a lot more chickens used to get run over on roads or something, but I don't think this is a strong reading. Tons of other common animals would work better here, I think, if it were intended. Even cows, if we want to stick to livestock, as I've seen or heard-of way more cows obliterated on roads than chickens... not sure I've ever seen a dead chicken on a road, in fact, and I've lived years in various parts of farm country before, had a grandmother who kept chickens that ranged rather freely, and keep chickens myself. I expect the most common cause of chicken-roadkill would be from trucks full of chickens crashing or losing some of their cargo, not from a loose chicken crossing a road like a deer or dog (or cow, or fox, or skunk, or armadillo, or various non-chicken birds, or cat, or bobcat, or possum, or any of the other animals I've seen plenty of dead on roads, which animals none of them were chickens). In fact, I'd go a little further and say sense 1 works as well as it does in part because chickens tend not to cross roads so often, preferring familiar lawn-space near their coop, which sets one to wondering what has enticed a chicken to essay such a crossing which, though not unheard-of, does seem like it must have some compelling motivation.

I think the piece is arriving at this reading by a bit of free-association, not seriously advancing it as a sensible way to read the joke.