I would like to re-structure the statement that you mentioned which is:
This means that the state as consumed by your target component can have a wildly different structure from the state as it is stored on your store
You can say that the state consumed by your target component has a small portion of the state that is stored on the redux store. In other words, the state consumed by your component would be the sub-set of the state of the redux store.
As far as understanding the connect() method is concerned, it's fairly simple! connect() method has the power to add new props to your component and even override existing props. It is through this connect method that we can access the state of the redux store as well which is thrown to us by the Provider. A combination of which works in your favor and you get to add the state of your redux store to the props of your component.
Above is some theory and I would suggest you look at this video once to understand the syntax better.