The beneficial interest in a contract relates to the assets of other benefits granted to a given party by the terms outlined in the contract. This is in contrast with legal interest, which is solely limited to the legal rights one has in relation to a contract. As an example, in a contract between a service provider, a middle-man and a client, the service provider, who'll be receiving payment, has a beneficial interest in the contract - while the middle-man has a legal interest, in the sense that they're only obliged to oversee the completion of both sides' obligations to the contract.
"Beneficial interest" may be used as a term with a more broad definition too, relating to any interest a person has in property which they do not currently own, even when not related to a contract. For example, individuals may have a beneficial interest towards the assets of a given company, without being bound or entitled to those assets in any way.
In its broadest sense, beneficial interest is used to describe situations in which a party receives some form of benefits from participating in a contract, when those benefits aren't related to their original purpose for participation - for example, any additional assets acquired when purchasing a real estate property that aren't related to the property itself. It should be noted that while the rights of a trustee to the assets described in a will could be considered beneficial interest, they actually fall under another category and are treated separately.