The person must have something to gain by the couple not being together. Jealousy is usually the cause of a person trying to breakup a couple. If it is a parent, they might fear losing a close relationship with their child. If it is a friend, the reason could be the same (losing a friend). Another thing about parents is that they often think no one is good enough for their child, especially a daughter.
Another reason could be concern for one of the people involved in the relationship. If the person suspects, or has evidence that someone is cheating, lying or hiding something from the other half the person may try to give the "victimized party" a heads up. If a person suspects, mental, physical or emotional abuse is happening this might also motivate them.
Another option is that the person wants to see the couple apart so they can slide in and take the partner. Example: the person wants your man (or woman) so they work hard to get you out of the picture. Or maybe they want you, so they try to get your partner out of the picture.
To make a long story short, they are either concerned or they are jealous. And yes you are right, it is the couple's business if they want to be together, and yes if there is anything "going on" the truth usually comes out one way or another. Hearsay is not concrete evidence, it's just gossip if there's no proof.