Equitable interest is the right to buy that a signed purchase contract gives you. The seller has agreed to sell to you at a set price. You don't own the house, and you can't paint it or rent it or borrow against it, but you hold an enforceable claim to buy it on those terms. That claim has value, because if the agreed price is $180,000 and the house is worth $210,000 to a flipper, whoever holds the right to buy at $180,000 is holding something worth paying for.
So the thing you sell is the contract, and the payment is called an assignment fee. Your buyer steps into your shoes and closes with the seller directly. The person who told you it's basically the same as owning is wrong in the way that matters most: you have no title, no insurable interest in the building, and no ability to convey the property yourself.
Two things that follow from this and trip people up. Your contract has to actually permit assignment, so "and/or assigns" after your name or an explicit assignment clause, and some sellers and some bank-owned sales will strike that out. And several states now treat publicly advertising the house itself, rather than the contract, as unlicensed brokerage, with the specifics differing state to state. That's why experienced people are careful to say they have a contract for sale, never a house for sale. Which rules apply to you depends on your state, and an attorney there is the one to confirm it.