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This definition is a different way of stating the basic fact that state functions are properties that allow us to take the final, or the output, and subtract it from the initial, or the input, to get an answer that shows its difference. What the definition means is that it does not matter how the output was achieved because it does not change the final value of the state function, such as enthalpy. Hence a state function, like enthalpy, is "a property of a substance that is independent of how the sample was prepared," how the enthalpy change occurred because the final outcome will always be the same: heat loss and heat gain (energy loss and energy gain). State functions only depend on conditions at current equilibrium and not the path that was taken to arrive at the equilibrium. Hope this helped and I didn't confuse you more! :)
Since the path taken to reach the final state does not affect the property, the state function is independent of the path function. Therefore, if we are only concerned with state functions, they can be added and subtracted because we're only working with the one given state of the enthalpies.
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