How to tighten a carriage bolt?
Do not drill the holes bigger in the wood parts. The idea with carriage bolts is that you pound them in with a hammer so that the square part under the head of the bolt is forced down into the tight hole in the wood. This provides the hold to a) keep the bolt in place during assembly and b) to keep the bolt from turning while using a wrench or socket to tighten the nut from the end of the bolt.

Carriage bolts are designed so that on one side of an assembly that there are no protruding bits for things to catch on. The rounded head will provide that as opposed to there being a hex or square head sticking out. In your case with a chair kit the general idea is that the heads of the carriage bolts should face toward the side of the chair where the user will be. As such the threaded bolt ends will want to be down underneath or toward a generally non used surface of the chair.