Here's a data structure that represents a bill of materials. It has a list of components, and a total cost, which is the sum of the costs of the components on the list. That's the invariant - that the total cost is the sum of the costs of the components in the list.
If it's just a structure, then someone can add or remove a component, and forget to update the total cost, and the invariant is violated. But if it's an object, if the code is bound to the data (in the way OOP normally calls encapsulation), then only the member functions can modify the structure. So if you want to add a component, you have to use the object's addComponent method, which (in a properly debugged object) isn't going to forget to update the total cost.
Now, you could say that the structure could have a code library to go with it, and that could have an addComponent function. That's true. But the user doesn't have to use that function - they can mess with the list directly, if they think they know what they're doing. Whereas with OOP encapsulation, they have to use the existing function. That function can guarantee that the object's invariants are maintained.