1. The children made up such a believable story that their mother fell for it hook, line and sinker.
2. The salesman was such a smooth talker, he could make anyone believe his stories. People always swallowed them hook, line and sinker.

The expression originates from fishing. One expects the fish to bite only the hook, but in some cases the fish might be taken in so completely and foolishly that it swallows not only the hook but the fishing line and the sinker as well.